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ON EXACT SHEARING SOLUTIONS OF THE NONSTATIC SPHERICALLY
SYMMETRIC EINSTEIN FIELD EQUATIONS
Theador H.E. Biech
B.Sc., Simon Fraser University, 1979
M.Sc. Simon Fraser University, 1982
THESIS SUBMITTED IN PARTIAL FULFILLMENT OF
THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
in the Department
of
Mathematics and Statistics
0 Theador H. E. Biech 1988
SIMON FRASER UNIVERSITY
August 1988
All rights reserved. This work may not be reproduced in whole or in part, by photocopy
or other means, without permission of the author.
APPROVAL
Name: Theador H. E. Biech
Degree: Ph.D. Mathematics
Title of Thesis: On Exact Shearing Solutions of the Nonstatic
Spherically Symmetric Einstein Field Equations
Examining Committee:
Chairman: Dr. G , Bojadziev
Dr. A. Das Senior Supervisor
Dr. E. Pechlaner
Dr. C.Y. Shen
Dr. M. Singh
I '
Dr. ~i0.J. Tupper External Examiner Department of Mathematics and Statistics University of New Brunswick
Date Approved: August 5, 1988
PARTIAL COPYRIGHT LICENSE
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T i t l e o f Thesis/Project/Extended Essay
Author: J
(s ignature)
( name
(date)
ABSTRACT
In this thesis we have sought solutions of the nonstatic
spherically symmetric field equations which exhibit non-zero
shear. The Lorentzian warped product construction is used to
present the spherically symmetric metric tensor in double-null
coordinates. The field equations, kinematical quantities, and
Riemann invariants are computed for a perfect fluid stress-
energy tensor. For a special observer, one of the field
equations reduces to a form which admits wave-like solutions.
Assuming a functional relationship between the metric
coefficients, the remaining field equation becomes a second
order nonlinear differential equation which may be reduced to a
Bernoulli equation. Some special solutions are found which have
shear and satisfy various weak energy conditions.
The double null coordinates are also used to study the
existence of a timelike collineation vector parallel to the
velocity of an anisotropic fluid. The resulting solutions are
reducible spherically symmetric spaces.
iii
DEDICATION
I dedicate this thesis to my parents.
ACKNOWLEDGEMENTS
The author thanks his advisor, Professor A. Das, for his
constant support. He also thanks Professor C. C. Dyer of the
University of Toronto for access to the MuTensor and REDTEN
computer algebra systems.
TABLE OF CONTENTS
Approval .................................................. ii
List of Figures ........................................... ix I. Motivation and Historical Background of the Problem
Statement of Problem.............................7
Methodology..,.................. ................. 9 Survey of Recent-Research.., ,....................14
Nonstatic Spherically Symmetric Anisotropic
Fluids and Conformal Symmetries., ................ 22 Results and Conclusions.....,....................24
11. Manifolds, Tensors, and Lorentzian Warped Products
Introduction.........,......,., .................... 27 Vector Spaces and Tensor Algebra..........,......28
Differential Manifolds and Smooth Maps..... ...... 30 Differential Geometry ............................ 44 Tensor Analysis on Differentiable Manifolds,.,...47
Elementary Causality on Lorentzian Manifolds.....52
Lorentzian Warped Products.......................58
Elementary Causality of Lorentzian Warped Product
Manifolds of the First Type...... ................ 61 Spherically Symmetric Lorentzian Warped Product
Manifolds of the First Type... ................... 62 Observers and the Kinematics of their Motion.....67
Motions on a Lorentzian Spacetime ................ 74 111. The Mathematics of the Stress-Energy-Momentum Tensor
Classification of the Stress-Energy Tensor.......79
Energy Conditions................................81
Stress-Energy-Momentum Tensors...................84
IV. Shearing Perfect Fluid Solutions in Spherically
Warped Product Manifolds of the First Type
The Field Equations in NN-Coordinates,,..,.......88
Solutions of the Field Equations in
NN-Coordinates................,,............,......97,
V. Spherically Symmetric Anisotropic Fluids in
NN-Coordinates
Anisotropic Stress-Energy Tensors and Conformal
Collineations.....,..............................l32
The Anisotropic Field Equations in
NN-Coordinatese..................................l39
A Study of the Conformal Collineation Equations..l44
Appendices
A. Transforming Systems of Notation in General
Relati~ity.....,..~............,.,.............~.l64
B. Symbols and Notati0n.......,..........~,.....,....l68
C. Calculations in NN-Coordinates...................172
vii
D . A MuTensor Program for a Perfect Fluid in .................................. NN-Coordinates. 179
Bibliography .............................................. 185
viii
LIST OF FIGURES --
FIGURE
1
2
3
4
5
6
7
8
PAGE
The Coordinate Transition Maps. .......................A
A Smooth Map between Manifolds M and N........,......,33
Curves in a Manifold.............,,..................,34
................................ The Pullback of a Map 3 5
The Fibre Bundle Construction..........................39
Vector Field over a Mapping .............................
The Strengths of the Elementary Causality Conditions.,58
2 2 Functions Related to (ru/f )u = (rv/f )v.............lOO
CHAPTER I
MOTIVATION AND HISTORICAL BACKGROUND OF THE PROBLEM
Introduction
In classical general relativity there are several major
unresolved problems. Many of these problems involve the
phenomena of gravitational collapse. Gravitational collapse is
very important, since it simultaneously presents the greatest
prediction of classical general relativity, and poses the most
difficult problems of the theory. The most important unresolv-
ed problem is the cosmic censorship hypothesis which was first
posed by Penrose [I]. There are several versions of this
conjecture 123 but they all essentially assert that "realistic"
gravitational collapse will result in a singular final state in
which the singularity is enclosed in an event horizon. This
conjecture is important for several reasons [31:
(a) singularities cannot appear at random in spacetime
but are always enclosed in event horizons;
(b) important results in general relativity, such as the
Schoen-Yau Positive Mass Theorem [4,5], the Black
-Hole Area Theorem 161, and several others, assume
the validity of the cosmic censorship hypothesis;
(c) in astrophysics, the final states of stars whose mass
exceeds a certain threshold must be black holes and
not "naked" singularities.
There are many papers in the literature documenting the
attempts to resolve this conjecture. Much of this work has
been in the production of examples which test various aspects
of this problem. A common feature of many of these examples is
that they are physically unrealistic. An example is collapsing,
radiating dust conjured in a manner so that, as the singularity
forms, the matter becomes evanescent. This approach, which is
entirely mathematical, has few advantages from a physical point
of view. These examples also tend to neglect the effects that
realistic matter might impose on the collapse. Seifert [2,7]
has put forward the idea that the shear of a realistic collaps-
ing fluid, and hence the viscosity due to shear-stresses, may
be important for excluding naked singularities. Israel [8,9]
has put forth a weaker conjecture, called the event horizon
conjecture, for which there are no known counter-examples, The
event horizon conjecture asserts that, when gravitational
collapse has proceeded beyond a certain critical point, an
event horizon forms. The nature of the critical point has been
left vague. One could take Thorne's hoop conjecture [ l o ] as an
indicator of the critical point: "Horizons form when and only
when a mass M gets compacted into a region whose circumference
in every direction is 1: S ZZ(~GM/C~)." One might also
conjecture that the event horizon forms in order to preserve
the generalized second law of black hole thermodynamics. In
any case a deeper study of the properties of matter and its
motion during gravitational collapse will be necessary before
these issues may be resolved.
Gravitational collapse has been studied in several
contexts :
(a) cosmological studies of galaxy formation;
(b) astrophysical studies of supernovae;
(c) study of formation of white dwarf stars and neutron
stars ;
(d) study of the possible physical origin of singularities
of solutions of Einstein's field equations.
C.W. Misner [ I l l has distinguished three types of gravitational
collapse :
(a) stabilized collapse when matter coagulates to form
well-known astrophysical objects such as planets,
stars, and galaxies;
(b) catastrophic collapse when matter is in near free-fall
with increasing density and remains in this state;
(c) dynamical collapse when a catastrophic collapse is
terminated by the formation of a quasi-static central
mass such as a white dwarf star or a neutron star.
As gravitational collapse is associated with matter, we
should expect that the mathematical properties of the models
which we use for matter should appear prominently in the study
of gravitational collapse. It is an unfortunate aspect of the
Einstein field equations that they are so difficult to solve in
general. The specification of realistic models for matter
makes these equations even more intractable. We are forced
into using idealized matter models which lead to solvable
systems of partial differential equations. These simpler
models have their cost however.
Astrophysical objects are usually modelled in general
relativity as perfect fluids. Perfect fluids are fluids whose
internal resistance to flowing is zero. Realistic models of
astrophysical objects should include viscosity, heat flows, and
electromagnetic fields. For large classes of applications in
astrophysics, static perfect fluid models are adequate, but
recently anisotropic (the pressure is different in different
directions) fluids and nonstatic perfect fluids motions have
become topics of active research in general relativity. The
objective in these lines of research is to produce models for
the nonstatic interior of stars, compact ultradense objects,
and to study the evolution of radiating spheres and gravita-
tional collapse.
Viscous fluids are a specialization of anisotropic fluids
to the case where the shear tensor of the fluid velocity is
proportional to the anisotropic pressure. These fluids are of
interest since they have been used to model relativistic quasi-
static dissipative processes near thermodynamic equilibrium.
Cutler and Lindblom El21 have studied the effect of fluid
viscosity on neutron star oscillations. Some of their inter-
esting conclusions are that neutron star matter becomes more
viscous in the superfluid state and that the dominant energy
dissipation mechanism in neutron stars is the shear viscosity.
Exact solutions for viscous fluids are very difficult to
find directly from'the Einstein field equations. Nonstatic
exact viscous solutions could be used as realistic models of
gravitational collapse and might show some insight into the
problem of cosmic censorship. Some of the technical difficult-
ies of viscous gravitational collapse have recently been
discussed by Coley and Tupper [ 1 3 ] . The problem we propose to
study is related to, and was partially motivated by, the
problems pointed out by Coley and Tupper. Coley and Tupper
point out that that there are very few known viscous-fluid
solutions of the Einstein field equations and that the problem
of matching these solutions to a portion of the exterior
Schwarzschild solution is not possible in general. Coley and
Tupper also mention the difficulty in the matching problem
related to the choice of interior and exterior coordinate
systems. A problem which was not discussed by Coley and Tupper
is the choice of the thermodynamic theory to be used in these
studies. Usually the Eckart [I41 model of relativistic
thermodynamics is used for these studies. However, it suffers
from a serious defect in that it is an acausal theory in which
there may be certain signals which propagate with speed larger
than that of light. Israel 1151, and Niscock and Lindblom [I61
have extended the Eckart theory in ways that make it causal.
These new theories are much more complicated than that of
Eckart. To our knowledge there have been no attempts to use
these new thermodynamical theories to create realistic models
of gravitational collapse.
A large collection of solutions of the viscous Einstein
field equations is desirable so that the effects of viscosity
may be better understood. Any method which may either
reinterpret known metrics or which will help generate new
solutions will be of interest. Through the work of Tupper
[17,18,19] and others [20,21,22], it is well-known that a given
metric tensor does not lead to a unique physical stress-energy
tensor and thus may have various physical interpretations.
Tupper has given conditions that allow certain perfect fluid
solutions to be reinterpreted as viscous magnetohydrodynamic
fluid with heat conduction. In [18] Tupper shows how under
certain conditions that a perfect fluid spacetime may be re-
interpreted as a magnetohydrodynamic fluid. In [I91 Tupper
shows how to find, under appropriate conditions, a viscous
magnetohydrodynamic reinterpretation of a given perfect fluid
solution. Tupper's method depends on finding the fluid
velocity of the viscous magnetohydrodynamic fluid by equating
the stress-energy tensor of the perfect fluid solution to the
viscous magnetohydrodynamic stress-energy tensor. The
equations needed to find this velocity involve the shear of the
viscous magnetohydrodynamic fluid. The viscosity introduces
some freedom of choice for the viscous magnetohydrodynamic
velocity. Resolution of this problem raises interesting
questions about the inheritance of the symmetries of the
perfect fluid solution by the magnetohydrodynamic solution. A
mathematical description of the possible physical situations
that may arise through Tupper's method can be made using the
Raychaudhuri equation (see Chapter 2). Tupper's method has
also been applied in studies of various cosmological models
[23,24,25,26]. A large collection of perfect fluid solutions
will be needed for effective use of Tupper's method in
modelling realistic gravitational collapse.
As the collapse of matter is a nonstatic process, we
should try to find nonstatic perfect fluid solutions. The role
of shear has appeared repeatedly in the previous discussion
hence we will look for nonstatic spherically symmetric perfect
fluids with shear and possibly other nonzero kinematical
quantities.
Statement of Problem
The Einstein field equations inside matter are
(1.1) G ab = T a b ,
(we use units with 8nG = c = 1) with reasonable physical side
conditions on the stress-energy-momentum tensor Tab. The
system of equations (1.1) is a quasi-linear second order
coupled system of ten partial differential equations for the
ten unknown functions of the metric tensor gab. The reasonable
side conditions usually imposed on Tab are that it should
describe macroscopic matter with everywhere non-negative energy
density, nonspacelike momentum flows, and pressures rather than
tensions.
From (1.1) and the differential identity
we deduce that the stress-energy-momentum tensor must satisfy
the equations
In conjunction with these side conditions we may impose
conditions on the admissible matter motions. If we assume the
matter is modelled by a fluid we may insist that the flow of
the fluid has certain specified kinematical properties such as
nonzero shear, acceleration, vorticity, or expansion.
From the preceding section we expect that shear will play
a very important role in the study of realistic gravitational
collapse. It would be useful to extend the collection of non-
static spherically symmetric interior solutions with shear and
other kinematical properties, There are many matter models one
could choose for the interior of matter. We will concentrate
on perfect fluids and later discuss some results of recent work
on anisotropic fluids.
Kramer et al. 1 2 7 1 have reviewed the known (in 1980) non-
static spherically symmetric perfect fluid solutions. Later in
their plenary survey of exact solutions of Einstein's field
equations Kramer and Stephani [ 2 8 ] note that only a few
radially shearing solutions for perfect fluid interiors are
known. This general class of solutions should be very large.
Nonstatic perfect solutions can be classified according to
properties possessed by the fluid flow i.e. shear, acceler-
ation, vorticity, and expansion. All solutions in a spheric-
ally symmetric spacetime must have zero vorticity as the
direction of vorticity would select a preferred direction.
We shall consider the problem of finding exact nonstatic
spherically symmetric shearing solutions of the perfect fluid
field equations and of the anisotropic fluid equations. In
particular we want solutions that have flows which are shear-
ing, expanding (or contracting), and which are accelerating.
The known classes of these solutions with these properties are
very sparse due to the inordinate difficulty in solving the
field equations. We will not apply the causal thermodynamic
theories to our work.
Methodology
The physical method for finding exact interior solutions
in General Relativity has essentially three steps:
(1) selection of allowable symmetries;
(2) selection of a matter model; and
(3) selection of initial (boundary) conditions.
The selection of allowable symmetries will usually constrain
the mathematical form of the aetric tensor. Often there is
more than one system of coordinates which are "adapted" to the
symmetries, thus a choice of coordinates may be involved in
this step. The selection of a matter model not only may
include the type of material (dust, radiation, perfect fluid,
viscous fluid, anisotropic matter, etc.), but also may include
an equation of state or a choice of thermodynamical model. We
will neglect the contributions of other physical fields such as
the electromagnetic fields and neutrino fields.
The selection of boundary conditions may involve initial
conditions on a spacelike hypersurface, junction conditions on
a timelike hypersurface (this also requires a choice of
"exterior" spacetime model), or asymptotic conditions on the
"boundary" of spacetime.
Even while employing very special choices in the physical
method we will often encounter intractable systems of partial
differential equations. The literature abounds with variants
of the above strategy employed to overcome the practical
* difficulties encountered. In many cases numerical integration
is used to find approximate solutions which yield useful
information about the model.
An alternate method is to relax the "imposed" physics and
to adopt mathematical stratagems which will simplify the field
equations to the extent that exact solutions may be found. We
will call this strategy the method of mathematical simplicity.
The physical interpretation of the solutions found in this way
depends to a large degree on how much of the matter model has
been retained. Often an explicit equation of state is not
imposed which leads to the necessity of deriving it (if
possible) from the field equations and the twice contracted
Bianchi identities.
The mathematical conditions imposed usually fall into two
classes :
(a) special forms of the metric tensor, and
(b) invariance of the physical fields under special
motions or symmetries.
The two classes are not disjoint as special motions or
symmetries will in general constrain the form of the metric
tensor. Often some of the metric tensor components are assumed
to be separable or have special functional forms of the
coordinates. The invariance of the physical fields under
various motions include Killing vector fields, homothetic
motions, conformal Killing vector fields, affine conformal
collineations, and others [ 2 9 ] .
Treatment of perfect fluid models usually involve ad hoe
simplifications to obtain analytical solutions for the metric
coefficients, the pressure, and the density, These models are
only approximations of the real situation in nature. The
degree of unrealism in a model is a cost of the modelling
process that is hopefully compensated by simple analytic
solutions. Often the unrealism appears in the form of unusual
equations of state.
A vexing problem in these investigations is that a
solution which has a very simple appearance in a certain
coordinate system may have a very complicated appearance in
another coordinate system. Many times equivalent solutions
have been rediscovered by various researchers who were either
unaware of the other solutions or could not see the equi-
valence. There is an algorithm for determining the equivalence
of metrics but it is extremely difficult to carry out with hand
calculations. Recently researchers have claimed that computer
programs exist which can perform this difficult task.
The choice of a Lorentzian manifold as a model in which to
put spacetime physics reflects our desire that spacetime should
be a continuum (at least at the non-quantum level). A very
useful tool, when it can be employed, is the concept of
Lorentzian warped product manifolds. The notion of Lorentzian
warped product manifold seems to have been first studied in
General Relativity by Delsarte [301 in the 1930's. In the late
1960's Bishop and O'Niell [31] independently reintroduced the
notion when studying manifolds of negative sectional curvature.
Later O'Niell 1321 and Beem and Ehrlich [33] published books
which have popularized the notion of Lorentzian warped product
manifolds. Lorentzian warped product manifolds are extremely
useful for the study of elementary causality. The Lorentzian
warped product may be used to represent spherically symmetric
metrics.
An important step in the solution of the field equations
is the choice of coordinates. A suitably chosen system of
coordinates may greatly assist in the solution of a difficult
problem. On the other hand, a badly chosen system of coordin-
ates may render the problem completely intractable. We shall
employ double null coordinates for the reason that they make
partial integration of the field equations easier. The idea for
using double null coordinates comes from their use in the
Kruskal-Szekeres completion of the Schwarzschild solution.
Double null coordinates have been used before in the study of
spherically symmetric interior solutions. Buchdahl [34] uses
double-null coordinates in a search for the interior source of
the Kruskal solution. He is motivated in his work by the
earlier observation made by Synge [35] that by writing the
spherically symmetric metric in double null coordinates, one is
lead directly to the Kruskal solution for a vacuum. These two
papers, particularly that of Synge, have greatly influenced our
choice of double null coordinates for our investigation of
perfect fluids. There may be disadvantages for physical inter-
pretation of any solutions that we may find, but as Stephani
[36] has noted that there is no overabundance of exact spher-
ically symmetric perfect fluid solutions.
Survey of Recent Research
A literature survey on exact interior solutions in general
relativity, even when restricted to a particular form of the
stress-energy-momentum tensor, is a large task. In the survey
which follows we have tried to present as complete as possible
listing of the work in the last 20 years which has as its
primary objective the construction of exact nonstatic spher-
ically symmetric perfect fluid solutions. In particular we
have concentrated on those papers that have matter flows with
nonzero shear and nonzero pressure. The zero pressure "dust"
solutions are unrealistic in our opinion. Collins [37],
Collins and Wainwright 1381, and Ellis [39] have extensively
discussed the role of shear in cosmological and stellar models
in general relativity. We have included some of the more
important studies on shear-free solutions as well. There are
three groupings of papers depending upon whether the shear of
the nonstatic perfect fluid solution has or has not been
analyzed.
First we shall review some of the earlier work on
nonstatic perfect fluid solutions whose flows were
unclassified. Secondly we shall review some of the important
recent work on exact nonstatic perfect fluid solutions with
shear-free flows. Finally we shall review the recent work done
on exact nonstatic perfect fluid solutions with shear (and
perhaps other kinematical properties).
There are many nonstatic solutions whose perfect fluid
flows have been unanalyzed. A common technique of finding new
solutions is to place restrictions on the form of the metric.
McVittie [40] has found a class of nonstatic spherically sym-
metric perfect fluid solutions. McVittie assumes a particular
form of the spherically symmetric metric tensor which allows
the field equations to be written as a system of three ordinary
second order differential equations. It has become common to
call spherically symmetric metrics with the special form "McV-
metrics". A property of the McV-metrics is that they depend on
only one function of the timelike coordinate. This property is
useful for seeing if a given metric belongs to the McV-metrics.
In a later paper McVittie [41] elaborates on his method and
relates the work of many researchers to his results. Dyer,
McVittie, and Oates [42] have examined the connection between
McVittie's method and the hypothesis that the metric tensor
admit a conformal Killing vector. Dyer et al. find conditions
that will force McV-metrics to admit conformal Killing vectors
but they are not able to resolve the issue of whether this
symmetry follows from the special form of the metric or the
assumption of a similarity variable that McVittie uses in
motivating his form of the metric. The investigations of Dyer
et al. were motivated by the results of Cahill and Taub 2431
who found that the existence of a homothetic Killing vector
implies the existence of a similarity variable. Cahill and
Taub studied the problem of finding spherically symmetric
perfect fluid similarity solutions.
Bonnor and Faulkes 1441 found a class of nonstatic
spherically symmetric perfect fluid solutions of uniform
density but nonuniform pressure. McVittie [41] has shown that
the class of solutions found by Bonnor and Faulkes is a special
case of a McV-metric. The motion of the fluid was unanalyzed.
Thompson and Whitrow [45,46] studied nonstatic spherically
symmetric perfect fluid bodies under the hypothesis that the
density is uniform. Using a simple mathematical condition on
the metric and imposing regularity on the solutions, Thompson
and Whitrow were able to prove a theorem that showed the
perfect fluid motion must be shear-free.
Nariai [47] studied gravitational collapse of a perfect
fluid with a pressure gradient. The solutions found by Nariai
are a special case of McV-solutions [41].
Faulkes [48] investigated nonstatic perfect fluid spheres
with a pressure gradient. Faulkes solutions are special cases
of those of Nariai [47] hence are McV-metrics as well. The
kinematics of the fluid were unanalyzed.
Eisenstaedt 1491 studied applications of perfect fluids to
cosmology under the added hypotheses of a barotropic equation
of state and uniform density.
In a long series of recent papers Knutsen [50,51,52,53,54,
55,56,57,58,59] has studied properties of nonstatic perfect
fluid spheres. In most of his work he uses either the McV-form
or a "generalized McV-form" of the spherically symmetric
metric. Knutsen has found nonstatic analytic models of gaseous
spheres (pressure and density are zero on the boundary). The
kinematics of the fluid motion are unanalyzed for the solutions
he finds.
Most of the known perfect fluid solutions have zero shear.
The earliest nonstatic spherically symmetric shear-free perfect
solutions were found by Wyman [60,61] but he did not recognize
them as such 1371. Many nonstatic spherically symmetric
shear-free perfect fluid solutions have been found by
Kustaanheimo and Qvist [62,63]. A general class of solutions
which are shear-free but expanding are contained in the class
of Kustaanheimo and Qvist. Many special cases of this general
class have been rediscovered by other researchers [271.
Stephani [36] has found a new class of shear-free perfect fluid
nonstatic solutions. Stephani uses the method of Kustaanheimo
and Qvist to find solutions that were overlooked in the earlier
paper. Banerjee and Banerji [64] have studied perfect fluids
with nonuniform density and pressure distributions under the
assumption of shear-free radial motion, Glass [65] uses the
method of Kustaanheimo and Qvist to study shear-free
gravitational collapse. Glass studied a particular second
order nonlinear differential equation earlier considered by
Faulkes [48]. Glass finds several shear-free collapsing
solutions. McVittie 1411 shows that some of the solutions of
Glass may be represented as McV-metrics. Glass [65] does
present solutions which are not McVittie metrics since the
solutions depend on two arbitrary functions of time.
Recently, Sussman 1661 has extensively surveyed the
spherically symmetric shear-free perfect fluid solutions (both
electrically neutral and electrically charged). Sussman finds
a large class of "Charged Kustaanheimo-Qvist" solutions
depending on two arbitrary functions and five parameters. The
neutral members of this class with special values of the other
parameters are identical with a class of metrics found by
McVittie [67]. In a later paper 1681, Sussman continues his
work on spherically symmetric shear-free perfect fluid to
examine the equations of state and singularities of these
solutions.
Srivastava 1691 has studied the methods of Kustaanheimo
and Qvist 162,631, McVittie [41], Nariai 1471, and Wyman [601
in an effort to achieve some degree of unification. Srivastava
finds that the McV-metric can be viewed as special case of the
Kustaanheimo and Qvist solution. Srivastava also shows that
the "generalized McV-metrics" used by Nariai [47] and Knutsen
1511 are equivalent to McV-metrics.
For the case when shear is nonzero there are only a few
solutions known. Recently it has been shown that some of the
previously known solutions with shear that were thought to be
distinct are the same.
Misra and Srivastava [701 has proven a stronger version of
the theorem of Thompson and Whitrow [45,461. Misra and
Srivastava showed that the mathematical condition of Thompson
and Whitrow may be dropped: All regular, nonstatic, uniform
density, perfect fluid solutions in comoving coordinates are
shear-free.
Some solutions with nonzero shear, but with some of the
other kinematical quantities (expansion, acceleration) equal to
zero, have been found. McVittie and Wiltshire [71] found a
special class of nonstatic shearing perfect fluid solutions
which exhibit no acceleration. McVittie and Wiltshire study
the use of non-comoving coordinates to find perfect fluid
solutions. There are also solutions of McVittie and Wiltshire
that have nonzero acceleration and expansion. Kramer et al,
[27] assert that Skripkin [72] has found a special class of
solutions with shear but zero expansion and constant density.
Under the hypothesis that the heat flux vanishes and
separability of the metric, Lake [731 found a class of general
fluid solutions with shear and with vanishing shear-viscosity,
A special case of his solutions reduces to that of Gutman and
Bespal'ko [74]. Shaver and Lake [75] have recently extended
the study of the solutions of Lake. Subject to the vanishing
of the heat flux they find that all such solutions with shear
and non-vanishing shear viscosity have a scalar polynomial
singularity at the origin. They conclude that for their form
of the metric the only fluid solutions of the field equations
with vanishing heat flux which satisfies the energy conditions
and are nonsingular at the origin are the Robertson-Walker
solutions.
In Kramer et al. [ 2 7 ] , a class of solutions with shear,
acceleration, and expansion were credited to Gutman and
Bespal'ko 1741 . (Note that this reference is in Russian and is
difficult to obtain). This class of solutions was found in a
comoving system of coordinates with the condition that one of
the metric functions was separable.
Vaidya [ 7 6 ] found a class df nonstatic solutions for a
perfect fluid whose streamlines are orthogonal to the isobaric
surfaces. These solutions have nonzero shear, acceleration,
and expansion.
Wesson [ 7 7 ] found a class of spherically symmetric
nonstatic solutions with shear with a stiff equation of state.
Kramer et al. [ 2 7 1 assert that Wesson's solution also has
acceleration and shear,
Van Den Bergh and Wils [ 7 8 ] have found three new classes
of exact spherically symmetric perfect solutions with shear and
acceleration and an equation of state. Assuming the ansatz
metric coefficients were that the
Wils als
separable Van Den Bergh and
o found a generalization of the stiff equation of state
solution of Wesson. This new solution has shear, acceleration,
and expansion.
Hajj-Boutros [79] has obtained several classes of non-
static, spherically symmetric perfect fluid solutions which
exhibit shear, acceleration, and expansion. One of the classes
is expressed in terms of Painleve's third transcendent
[80,81,82].
Collins and Lang [83] have recently found a class of
spherically symmetric spacetimes which exhibit shear and
acceleration. Collins and Lang imposed the condition of self
similarity on Lorentzian spaces with a perfect fluid. They
discuss the work of Gutman and Bespal'ko [74], Wesson f771,
Hajj-Boutros [79], Van den Bergh and Wils 1781, and Herrera and
Ponce de Leon [84]. Collins and Lang show that the metric of
Gutman and Bespal'ko [74] is the same as that of Wesson [77].
Collins and Lang also show that the metrics of Van den Bergh
and Wils [78] specialize to those of Wesson [77] and Gutman and
Bespal'ko [74]. Collins and Lang point out that some of the
work of Herrera and Ponce de Leon [84] with conformal Killing
vectors may be specialized to self-similarity and hence arrive
at the metric of Van den Bergh and Wils [78].
Nonstatic Spherically Symmetric Anisotropic Fluids
Conformal Symmetries
Spherically symmetric anisotropic fluids admitting
conformal symmetries have recently been studied. The reason
for this interest is that a connection between the existence of
conformal Killing vectors and the equation of state has been
found 1851 by Herrera et al. The metrics considered were
static and had a conformal Killing vector orthogonal to the
fluid velocity. The existence of a conformal Killing vector
was shown to constrain the pressure and density. It was also
found that the orthogonality condition together with a special
conformal Killing vector forced the fluid to have a stiff
equation of state. In [ 8 5 ] it is asserted that recent real-
istic studies of stellar models indicate that an anisotropic
fluid model may be more appropriate.
Herrera and Ponce de Leon 186,871 find families of expand-
ing/contracting fluid anisotropic spheres, and confined non-
static spheres whose total gravitational mass is zero. The
problem of matching these spheres to exterior vacuum metrics is
studied. These results have stimulated work on more general
classes of symmetries such as collineations [29,88,89].
Duggal and Sharma [90] have recently investigated the
similar dynamic restrictions imposed by a special conformal
collineation in a class of anisotropic relativistic fluids
without heat flux, In particular they showed that the stiff
equation of state is no longer singled out when the collinea-
tion vector is orthogonal or parallel to the velocity
vector.
Shortly thereafter Maartens and Mason [89] extended and
corrected some of the results of Duggal and Sharma. They
showed that the assumption, by Duggal and Sharma, that a
special conformal collineation vector preserves the fluid flow,
is equivalent to assuming that the velocity vector is an
eigenvector of the conformal collineation tensor (see Chapter 2
for the definition of the collineation tensor). Later Duggal
[91,92] points out that, for fluid spacetimes, conformal
symmetry plays the role of preserving the continuity of the
matter flow at critical points of transition during a change of
state. These ideas may have some use in cosmology. Duggal has
found a connection between the shear of the fluid and the
existence of a timelike conformal collineation vector. Duggal
is able to generalize the theorem of Oliver and Davis 1933 on
the existence of timelike conformal Killing vectors to the case
of timelike conformal collineation vectors. In his generaliz-
ation of the Oliver-Davis theorem, Duggal shows that a fluid
spacetime admits a timelike conformal Killing vector parallel
to the fluid velocity if the shear tensor is proportional to
the collineation tensor.
Duggal's theorem promises to be an important advance in
the study of shearing fluids. Duggal points out that it is an
extremely difficult problem to characterize the collineation
tensor so that physical examples of conformal collineation
vectors may be found. In Chapter 5 we shall collect some
partial results concerning this problem. In Chapter 5 and in
an appendix we shall calculate the field equations in double
null coordinates for an anisotropic fluid admitting various
types of collineation vectors.
Results gmJ Conclusions
In chapter 2 we present a summary of the mathematics
which is used in general relativity. The usefulness of the
Lorentzian warped product construction is noted, particularly
in relation to the problem of discussing elementary causality.
A specialized form of spherically symmetric double null
coordinates is introduced. These coordinates have the
advantage that they do not change their causal type when the
metric coefficient change sign. However they do not cover as
much of spacetime as the usual double null coordinates. A
short presentation of the kinematics of an observer concludes
the chapter.
In Chapter 3 we give a brief discussion of the
classification of stress-energy tensors and the energy
conditions which apply to them.
In Chapter 4 we use the double null coordinates to look
for solutions of the perfect fluid field equations which
exhibit shear.
The field equations lead to a "wave-like" equation and the
pressure isotropy equation. Although the wavelike equation
admits a first integral, it cannot be completely integrated in
general. The field equations were simplified by assuming a
functional relationship between the metric coefficients. With a
a functions relationship of the form f = r , was used for several
cases. In particular a = -2, -1, -1/2, 0, 1/3 lead to reason-
able solutions which are nonstatic and have shear. Another
a r case considered was f = e . The functional relationship
reduces the field equations to a Bernoulli equation. In six of
the cases, the timelike weak energy conditions were satisfied.
The dominant energy conditions were verified for four of the
solutions and the strong energy conditions for four of the
solutions. Unfortunately only two of the Bernoulli equations
was integrable in closed form in terms of elementary functions,
This fact greatly hindered analysis, particularly of the
causality, which had been planned. In spite of this, some
useful information was drawn from the mathematical form of the
Bernoulli equations and the energy conditions. The kinematical
quantities were computed when possible. Invariants derived
from the Riemann tensor such as the Ricci scalar and the
Kretschmann scalar were also computed. For all cases the
acceleration of the fluid is zero. The shear tensor was always
nonzero. An interesting observation about the solutions is
that they all satisfy the condition of being in a T-region
[ 2 7 1 . There have been very few studies of T-models, until now
only two or three solutions for dust, a stiff fluid, and a
radiation solution. Three of our solutions are not among these
solutions for the simple reason that they have much more
complicated equations of state than previously known solutions.
We found two solutions which have the equation of state
p = ~ / 3 .
In Chapter 5 we present some recent results by Duggal
connecting timelike collineations to the shear of a fluid.
Calculations for timelike collineations in double null
coordinates are presented. We state a theorem showing how a
stress-energy tensor may be used to try to find a coll'ineation.
A theorem asserting the existence of a timelike collineation
parallel to the velocity when given a collineation tensor and
the velocity field. The equations for an anisotropic fluid
admitting a timelike collineation vector parallel to the
generalized comoving velocity are written in NN-coordinates.
The condition that the collineation tensor is covariantly
constant is investigated in detail for this case. Two
solutions of the field equations result as a consequence of
this study. Both of the solutions lead to reducible spaces in
accordance with recent results of other researchers.
CHAPTER
MANIFOLDS, TENSORS, AND LORENTZIAN WARPED PRODUCTS
Introduction
In this chapter we shall present several sections summar-
izing the mathematical tools and notation used in this work.
This summary is a condensation of the treatments of similar
material found in the texts of Sachs and Wu [94], Hawking and
Ellis 1953, Wald [96], Stephani [97], Frankel 1981, O'Niell
[ 3 2 ] , Hicks [99], and Beem and Ehrlich [33]. Relevant comments
have been added to aid in the assimilation of the large number
of well-known definitions and concepts. More complete and
detailed discussions of these topics may be found in the cited
sources. In some cases there is still some disparity in the
literature on the use of certain terms and definitions and the
notation used to describe them. There is also a recognized
profusion of sign disparities among relativists and differ-
ential geometers in the basic definitions of the classical
tensor calculus. The uncertainty that these disparities impart
leads inexorably to qualitative errors, particularly in
expressions that involve the use of inequalities. We have
tried to eliminate the possibility of such errors by system-
atically classifying the notation systems and conventions used
by various authors. Appendix A is a summary of a system for
translating tensorial expressions from one system of tensor
calculations to another.
Vector Spaces Tensor Algebra
Let V denote a vector space over the real field R. We
shall almost always have dimpZV = 4 but most of the results are
true as long as dimRV is finite. The dual space of V is
denoted by v*. The underlying set of V* is the set of all
R-linear functional9 on V. V* is a real vector space and if
dimRV is finite then dimRv* = dimRV. If V and W are finite-
dimensional real vector spaces then V $ W denotes their direct
sum. V* b W* will denote the vector space of R-bilinear maps
V x W -----+ R. A standard theorem of linear algebra for finite
dimensional vector spaces states
Since dimRV is always assumed to be finite we have
where vX* is the dual space of v*. Another well-known theorem
of linear algebra [I001 then states that V st v**. This natural
isomorphism is used repeatedly in tensor algebra to eliminate
the unnecessary distinction of V from v**.
Associated with V are its tensor spaces of type (r,s),
T'(v), where (r,s) is a pair of non-negative integers. A s
tensor of type (r,s) on is defined to be a real-valued
multilinear functional on (v*)' x (V) '. A standard theorem of
tensor algebra asserts that dimR(~:(v) ) = [dimR(V) 1 '+'.
Tensors of type (r,O) are called contravariant tensors, while
tensors of type (0,s) are called oovariant tensors. T:(v) ray
be viewed as the real vector space of multilinear functionals
on r+s factors, the first r factors being v*, the remaining s
factor being V. The tensor algebra on V is the direct sum over
all pairs of non-negative integers (r,s) of the tensor spaces
r mOs=O
If V is a vector space over R, then an inner product on V,
g : V x V -----, R, is a symmetric, bilinear functional. The
inner product is the natural generalization of the dot product
on Rn. The index of g, Ind(g), is defined to be the maximum
dimension of the subspaces of V on which the restriction of g
is negative definite i.e.
(2.4) Ind(g) max{dimRW : W is a subspace of V}.
The nullity of g, N(g), is defined to be the dimension of the
subspace N of V on which the restriction of g is identically
zero i.e.
(2.5) N = {X E V : g(X,X) = 0 )
and
(2-6) N(g) 3 dimRN.
The signature of g, sig(g), is defined as
(2.7) sig(g) = dim# - 2Ind(g) - N(g). The Lorentzian inner products to be introduced later will have
signature +2 since dim$ = 4, Ind(g) = 1, and N(g) = 0.
Differential Manifolds Smooth Maps
Differential manifolds are the setting for the appropriate
generalization of calculus on vector spaces. Differential
manifolds provide a setting in which we may have the notions of
limit and derivative. Manifolds may be thought of as the
abstraction of the idea of a surface in Euclidean space. We
first introduce local objects (charts) on which limits and
differentiation make sense, and then we patch these objects
together in a smooth manner. Differential manifolds should not
be simply viewed as parameterized surfaces since the notion of
a parameterization involves the notion of an enveloping space.
The underlying set of a differential manifold is a
topological manifold. An n-dimensional topolojfical manifold is
defined as a separable Hausdorff topological M space such that
every point in M has an open neigbourhood which is homeomorphic
to an open subset of R", Note that M is a-compact,
paracompact, and has at most a denumerable number of components
(101,1021. The paracompactness of M is essential for the
theory of integration on manifolds. By considering simple
examples such as the sphere sn, n 2 1, and the torus T", n r 2,
in R"" we quickly see that it is impossible to coordinatize
these objects with a single coordinate chart. A similar
situation holds in many of the models of spacetime that we use
in general relativity.
A n-dimensional coordinate chart on M_ is a pair ( % , p a )
where Ua is an open subset of M and qa is a homeomorphism of Ua
onto an open subset of R" (a is an arbitrary index in the set
A; see Appendix B on symbols and notation conventions for the
rules which indices obey). For each integer i, 0 S is n-1, and
for each coordinate chart ( U a , p a ) we define the i-th coordinate
function of pa, xg, so that
(2.8) i i = n o c p a L : P1 -IR
a
where ni is the i-th canonical projection on I R n .
- -
Figure 1 The Coordinate Transition Maps
Since more that one chart will usually be necessary to
cover M, we must find a suitable compatibility criterion for
the overlap of the chart domains. Two charts ((11Q,(p~) and
(T6,q6) are 8'-compatible (see Figure 1) if the coordinate
transition maps
( 2 . 9 ) C
elu 'YeO~u : %CUu n 9-lgl 961% n P1,l
and
C (2.10) i ~P,OV, : n 91,l -
u 6 (pol% n U61
are fZr-diffe~mor~hisms (coordinate dif feomorphisms) between
their domains and ranges in R". (Note that p: denotes the
inverse mapping of IDQ.)
An indexed collection 4 = { ( Uu, (pot) : a E A) of charts
which cover M and satisfy the 8'-compatibility criterion for
all index pairs (a,#?) E A x A will be called a er-subatlas for
M. Two fZr- subatlases d and 3 of M are equivalent if d u 3 is
a 8'-subatlas for M. An equivalence class of 8'-subatlases for
M will be called a B'-differentiable structure for M. Using
inclusion as a partial order on a %'-differentiable structure
we define a 8'-atlas of M as the maximal element in the
e'-differentiable structure. It is easily shown using Zorn's
Lemma that a maximal atlas always exists. We shall always
assume that r 2 3. A 8'-differentiable structure always
a, contains a 8-structure thus we will simply use the adjective
"smooth" to denote the appropriate degree of differentiability.
Let M be a manifold endowed with a smooth differentiable
structure characterized by a subatlas d = { (Ua, cpat) : a E A ) .
For a coordinate chart (Uu,pa) E d and a function f : M IR,
a coordinate expression for f in (Uu,cpo), is the function
(2.11) fu 9 fopu : pu(Uu) c R" - R. A function f : M - R is smooth if fu is smooth in the
usual sense for all charts ( % , ( p ~ ) E d . We will use 9(M) to
denote the commutative ring of smooth functions.
- --
Figure 2 A Smooth Map between Manifolds M and N
Let N be another manifold with a smooth differentiable
structure given by the subatlas 93 = { (%,ly,) : @ E B) . A
3 3
continuous map F : M ----+ N will be called smooth if for each
index pair (a,p) E A x B the map C
( 2 . 1 2 ) B F a = : P ~ [ u ~ n F-'w$'B,I - VJFWJ n 51 (see Figure 2 ) is smooth in the usual sense. Y?(M;N) will
denote the set of smooth maps from M to N.
Figure 3 Curves in a Manifold
*, An important class of maps is f2 (I;M), where I is an open
interval in R. These maps are called curves in M. Let
0 E ~?(I;M) with a ( 0 ) = p, then, for each a E A such that
p E %, there is a local coordinate representation of U, (2.13) 4 3 puoa : (-E,E) c I -- q u ( k ) n R"
Figure 3) . almost never specify the curve a 4
Note that in practice we
3irectly. If a chart ( x , ' ~ , ) is given, we just specify the functions oa directly.
Figure 4 The Pullback of a Map CO
Each smooth map F € f3 (M;N) determines an algebra homo-
morphism F* : F(N) --+ 9(M) given by FX(f) = Fof where
f E S(N). The map F* will be called the pullback E (see
Figure 4).
A diffeomorphism F : M ---4 N is a smooth mapping that has
C an inverse mapping F : N ---4 N which is also smooth. The
theory of differentiable manifolds can simply be described as
the study of objects preserved by diffeomorphisms.
The main.step in extending the calculus from vector spaces
to differentiable manifolds is finding a generalization of
directional derivatives. The appropriate concept for a differ-
entiable manifold M is the tangent vector V at a point p € M. P
There are four different ways of viewing the notion of tangent
vectors on a manifold M [103]. Vectors at p E M may be viewed
as equivalence classes of curves through p, as equivalence
classes of n-tuples.of real numbers at p, or as derivations on
germs of functions defined on a neighbourhood of p. We will
adopt the view that tangent vectors are derivations on 9(M). A
tanaent vector of M & E is a linear map V : 9(M) - If? such P
that
(2.14) v [fgl = v [~IB(P) f (p)vp[gl P P-
for arbitrary functions f, g in F(M). The collection of all
tangent vectors at a point p E M is denoted T M, the tangent P
space to M E. We can define addition of tangent vectors at
(2.15) (V t W )[f] = V [f] t Wp[f]. P P P
and we can define scalar multiplication of tangent vectors at p
by
(2.16) (cV ) [f] = cVp[f]. P
It can easily be shown, using the operations above, that T M is P
a real vector space and dimR(T M) = dimR(M). For a given P
coordinate chart (flu,%) we can find an induced coordinate
(holonomic) basis for T M i.e. P
a : 0 * i S 3 . For a function f E 9(M) and a chart
containing p, U , ) , we define the action of aui 1 on f E +(M) P
4 where the ui are coordinates on R . For a smooth mapping F : M -----+ N and for each p E M we
define the map F : T M ----+ * P P TF(plN by
( 2 . 1 8 ) F ( V )[f] a V p [ F o f ] . *P P
By choosing holonomic bases for both T M, TFcp,N associated P
with chart ( respectively i. e e
( 2 . 1 9 )
and
( 2 . 2 0 )
we can find a coordinate presentation of F : i *P
( 2 . 2 1 ) eFUxp ( a u j I , ; F B O F ) ( P ) ] ~ J i I ~ ( p ) '
Xu
The term in the square brackets in ( 2 . 2 1 ) is the Jacobian
matrix of If 6Fa*p is injective then F is called an
immersion. If F is an injective immersion, with N homeomorphic
to F ( N ) , then F is an imbedding of N into M. A submersion
F : M - N is a smooth mapping onto N such that is onto
for all p E M . When the charts (Uu,pu) and (r6 , tp6) are fixed
in an argument we will dispense with the more complicated
notation pFuxp and simply write F . This same abuse of *P
notation will hold in other similar notational situations.
A submanifold N of M is a topological subspace of M such
that the inclusion map jN : N - M is smooth and for each n E N the induced map jNsn : T N --r TnM is injective (one-to- n
one). The submanifold N inherits a differentiable structure S N
related in a natural way to that of M. The notions of imbedding
and submanifold are closely related: if N is a submanifold of M
then the inclusion map jN is an imbedding.
An abstract construction which appears frequently in
theoretical physics and differential geometry is the fibre
bundle [lo21 construction. Locally fibre bundles are product
manifolds but the submersion ll may "twist" the product. Let a
smooth submersion ll : E ----+ B be given for smooth manifolds E
and B. The map n has the local product property (see Figure 5 )
with respect to a smooth manifold F if there is
(a) an open covering {% : a E A) of B, and (b) a family of diffeomorphisms {vu : 11 u x F --r n-'(q)l
such that Ilovu(b,f) = b for all b E Uu, f E F. (ll-' is
the usual inverse set function.)
The collection { (Uu,pu) : a E A ) is called a local
decomposition of ll. A smooth fibre bundle is a four-tuple
(E,n,B,F), where E is the bundle manifold, ll : E - B is a smooth submersion, B is the base manifold, F is a smooth
manifold called the typical fibre.
Figure 5 The Fibre Bundle Construction
There is a large variety of subtypes of the fibre bundle
construction, depending on the type of fibre and its internal
symmetry groups, but the one of interest is the vector bundle
construction. A vector bundle is simply a fibre bundle that
attaches vector spaces to each point of the base manifold B.
Many of the concepts and objects used in general relativity
have a natural presentation in the setting of vector bundles.
Let TM = {(p,V) : p € M, V E T M). The tangent bundle on P
PI, TM, is a special case of the vector bundle construction on M
[102]. TM is a smooth manifold with a smooth surjection
ll : TM -----, M defined by
(2.22) Np,V) = p.
Viewed as a fibre bundle, the base manifold of TM is M and the
fibre manifold at p € M is the vector space T M. An atlas on M P
induces a differentiable structure on TM in a natural way. If
((llQ,pa) is a coordinate chart on M then there is a natural
induced chart (?la x R ~ , pa pa*) on TM such that
where
(2.24) Il I v : ~ i ~ p P
is the representation of V with respect to the holonomic basis P
induced on T M by the coordinate chart (pa,Ua). The fibre P
Il-l(p) is isomorphic to T M and hence is identified with T M. P P
There are several other bundles of interest on M such as the
cotangent, tensor, exterior algebra, and frame bundles [102].
Of these the most important is the tensor bundle, T'M, of type 8
(r,s) tensors over M.
A vector field on a smooth map F : N -----+ M is a smooth
mapping V : N - TM such that lloV = F, where ll is the
projection TM ---+ M (see Figure 6 ) . There are three cases of
F that occur repeatedly in general relativity:
(a) F = a , where n : U c R --+ M is a smooth curve in M;
(b) F = j,, where jN : N -4 M is an imbedding of a smooth
submanifold N into M;
( c ) F = ' 'M ' where in is the identity map on M.
Figure 6 Vector Field over a Mapping
A vector field V over iM on a differentiable manifold M
may also be viewed as a function that assigns to each point
p E M a tangent vector V E T M. For a smooth function P P
f E 9(M), Vf is a function such that
(2.25) (Vf )(P) = VpWI
The vector field V is smooth if Vf is smooth for all f E 9 ( M ) .
It can be shown that the set of all smooth vector fields on M,
Z(M), is an S(M)-module [32]. The set of all smooth vector
fields over in may also be viewed as a vector bundle over in.
Thus a smooth vector field is a section of a particular vector
bundle. Let T be a tensor field over F. A derivation of
tensor fields over F is an assignment of a tensor field DT over
F such that
(a) DT is the same type of tensor as T;
(b) D(aS + /3T) = aDS + BI)T for all a,@ E I?, and all tensor
fields S, T over F;
(c) D(SW) = DSW + S@DT for all tensor fields S, T over
F.
A connection Q over F is an assignment to each vector field X
over F of a derivation Dx of tensor fields over F so that
(a) Dxf = Xf if f is a function over F;
(b) D*x+gu T = fDxT + gDyT for smooth functions f, g over
F;
(c) Dx commutes with contraction.
If F : N - M is a smooth map and D is a connection on M then F*D is the unique connection over F such that
(2.26) (F*D)~(VOF)(P) = D v(F(~)) F*X
for all vectors X E N and for all V E X(M). The connection P
F*D is called the induced connection.
The bracket of two vector fields V and W, denoted [V,W], is the
vector field such that for all p E M and all f E F(M) we have
The bracket operation on Z(M) has the following properties [32]
(a) [aV + bW,Xl = a[V,XI + b[W,X] for all a, b E R;
Vector fields are first order operators on functions in 9(M).
A surprising fact is that the bracket of two vector fields is
not a second order operator but is another vector field.
A smooth curve CJ : (-B,E) c I - M is a local integral curve of V E X(M) if a' . a*(ay) = (Voa)(u) for all values
u E - , c I. Occasionally the domain of a local integral
curve can be extended to the whole real line. If dom a = R and
a' = V then we say that V is complete. By considering the
smooth vector field V = ( 1 ,-y2) on IR2 it is easy to see that
not all smooth vector fields are complete even when dom(V) = M.
This arises since we have only a local existence and uniqueness
theorem for differential equations. For each p E M and
V E X(M) there is a unique maximal inte~ral curve in the sense
that the domain of a cannot be extended. For a vector field
V E X(M) we define a local flow as a map @ : U x I -----+ M such
that $(p,t) = 9 (t) where 9 (t) is the unique maximal integral P P
curve of V through p defined for all p E P1 and all t E I.
If F : M - N is a smooth mapping of manifolds then F induces an associated mapping between each of the tensor bundles T:M
and T'N. For type (1,O) we denote this map by TF and for type s
(0,l) we denote this map by TF*.
Differential Geometry
The setting for differential geometry is a manifold
endowed with a notion of "inner product" called a metric. We
wish to study the properties of manifolds and metrics which
remain invariant under a group of diffeomorphisms which
preserve some of the local algebraic properties (signature,
index, nondegeneracy, nullity, symmetry) of the metric.
Metrics are usually classified by their signature, nullity, and
index. The most general class of metrics is the class of
semi-riemannian metrics of arbitrary signature. The Riemannian
metrics and the Lorentzian metrics may be viewed as subclasses
of the semi-riemannian metrics.
A smooth assignment of a symmetric, nondegenerate bilinear
form g(p) : TpM x T M -- R such that the Ind(g(p)) = 0 and P
N(g(p)) = 0 for all p E M is called a Riemannian metric on M.
A manifold (M,g) such that g is a Riemannian metric for each
p E M is called a Riemannian manifold.
Similarly a smooth assignment of a symmetric,
nondegenerate bilinear form g(p) : T M x T M ----+ R with P P
arbitrary nullity is called a semi-riemannian metric on M. A
manifold (M,g) such that g is a semi-riemannian metric for each
p E M is called a semi-riemannian manifold,
As general relativity uses an inner product of index 1 and
nullity 0, we shall restrict our description of differential
geometry to this case, A smooth assignment of a symmetric,
nondegenerate bilinear form g(p) : T M x T M ----+ IR with P P
Ind(g) = 1 at each point in M is called a Lorentzian metric on
M. The pair (T M, g(p)) is a Lorentzian vector space for each P
p E Me The corresponding manifold is called a Lorentzian
manifold.
If (M,g) is a semi-riemannian manifold then there is a
unique connection D over the identity 5 such that for all V,
D is called the Levi-Civita connection of (M,g). The same
properties hold for a Riemannian manifold. These properties
define the interaction of the metric g and the connection D in
such a way as to render D torsionless, Levi-Civita connections
may be characterized by the Koszul formula [33]:
(2.30) Zg(D,W,X) = V[g(W,X)I - g(V,[W,XI) + W[g(X,V)I
- s(W,CX,Vl) - X[e(V,W)I + g(xtCvtw1).
A vector field V is parallel with respect to g if DxV = 0 for
all smooth vector fields X. If V = u x ( d ) is the tangent u
vector field on a curve a and DVV = fV for some smooth function
f, then we say that a is a pre-geodesic. If DVV = 0 then u is
a geodesic. The distinction between a pre-geodesic and a
geodesic lies in their parameterizations. A pre-geodesic may
always be reparameterized to be a geodesic. The parameter-
ization that makes a pre-geodesic a geodesic is not unique.
Parameters of a geodesic are affinely related to each other.
A smooth mapping F : (M,g) ----+ (N, h) is called an
isometry if F is a diffeomorphism and g = FX. A submanifold N of (M,g) is nondenenerate if for each
p E N and nonzero V E T N there is a W E T N so that P P P P
j:g(~ ,W ) # 0, where j> is the pullback of the metric g via P P
jw : N ----r M. If j> is positive definite then N is a
spacelike submanifold of M; if jig has index 1 on T N for all P
p E N then N is a timelike submanifold of M; if jig is
degenerate then N is a null submanifold.
A submanifold N of (M,g) is geodesic at p E N if each
geodesic y of (M,g) with ~ ( 0 ) = p and y'(0) E T N is contained P
in N for some neighbourhood of P. N is totally geodesic if N
is geodesic for each p E N.
A submanifold P of (M,g) with dimension dim$ = dim# - 1 has a number of distinguished forms defined on it. These forms
are derived from the unit normal vector N by the following
formulae :
Tensor Analysis on Differentiable Manifolds
For any pair of nonnegative integers (r,s) # (0,O) and a
ring K, a K-multilinear function R : (v*) ' x V* -----+ K will be
called a K-tensor of type (r,s) over 1. In general relativity
there are two cases of interest:
(a) K = R and V = T M; P
(b) K = F(M) and V = f(M).
If U c M is an open set in a semi-riemannian manifold
(M,g) and R : U -----. T:M is a map such that l l o R = \ then R is a type (r,s) tensor field on Q. A tensor field of type (1,O)
is called a vector field, A tensor field of type (0,l) is
called a 1-form field. M has a Lorentzian metric g thus there 1 0 is an associated isomorphism between TOM and TIM given by
b 1 : TOM - T> so that
a : O S ~ S ~ In terms of components with respect to pa - ax:
i and its dual basis {ot + dxa : 0 i i S 3) we have gaijv: = vai. 0 1 There is also an isomorphism between TIM and TOM given by
so that
In terms of components we have gi'v = aj
just the usual index raising and index
i # b vu. Thus and are
lowering isomorphisms of
classical tensor analysis. We see that the index raising
(lowering) isomorphisms define an equivalence between 1-forms
and vector fields. A 1-form o is metrically equivalent to a
vector field W if o' = W and wb = o.
It is well-known that tensor analysis can be done with
respect to arbitrary bases of T M and T*M. As all the tensor P P
analysis that we shall use is done with respect to holonomic
bases, we shall omit the extension of our notation to this
case. In many cases there is only one coordinate chart ( U u , ~ )
in use at one time so we will dispense with the subscript a
indicating which chart we are using. In situations in which
more than one chart is being used we will revert to the more
complex, but mathematically unambiguous, notation.
The Christoffel symbols of the second kind associated with
a metric tensor g and coordinates (\,pu) are written as
where gil is found from the equation
Covariant differentiation in a chart (U=,(P,) will simply
be denoted by appending a covariant index preceded by a
semicolon. Thus the partial covariant derivative of a tensor
In the formula ( 2 . 3 6 ) above the ";kt ' denotes the partial
covariant derivative with respect to auk and a comma denotes
the partial coordinate derivative with respect to xi.
The failure of partial covariant differentiation to
commute leads to the Ricci identities which may be taken as the
definition of the Riemann tensor:
The components of the Riemann tensor with respect to the
coordinate basis and the dual basis of the chart ( % , p a ) may be
written in terms of the Christoffel symbols of the second kind
as:
R~ jkl = ri - ri jk, jl,k + ri n 1 rnjk - rinkrnjl.
The Riemann tensor satisfies the following symmetry properties:
(2.41) - (C) Rijkl - Rklijt
(2.42) (d) R i ~ j k l ~ = 0,
where indices inclosed in brackets indicate normalized anti-
symmetrization. Contracting on the first and third indices
leads us to the expression
coordinates:
(2.43) R = Ra i j iaj*
for the Ricci tensor in the same
The Einstein tensor is defined by
where R is the curvature scalar given by
(2.45) R = R = . a
The Riemann tensor also obeys the well-known Bianchi
differential identities:
A contraction on the Bianchi identities leads to the first-
contracted Bianchi identities:
Ri jkl; i + 2RjIk;11
0.
Another contraction of the Bianchi identities implies the well-
known differential identities for the Einstein tensor:
(2.48) Gij = 0 . ; j
The Weyl conformal cu.rvature tensor is defined by
decomposing the Riemann tensor into products of the metric
tensor, the Ricci tensor, and the curvature scalar. The Weyl
conformal curvature tensor can be characterized by the fact
that it has all the symmetries of the Riemann tensor and is
traceless with respect to all contractions. Thus
and Weyl tensor satisfies the same symmetry properties as the
Riemann tensor. The Weyl tensor also is traceless:
In terms of the Weyl conformal curvature tensor it can be
shown [ 3 9 , 1 0 4 ] that for a 4-dimensional Lorentzian manifold the
Bianchi differential identities are equivalent to
( 2 . 5 1 ) C i j k l R k C i ; j l - k [ i R ; j l ( 1 /6 )g .
; 1
The Weyl conformal curvature tensor has a decomposition
for an observer with velocity u i in terms of its "electric"
part E and its "magnetic" part H i j given by [39]: i j
( 2 . 5 2 ) - 'i j k l - ( q i jpqqklrs + g i j p q g k l r s ) upurEqs
- ( q i j p q 8 k l r s +
where
( 2 . 5 3 ) j 1 ' ' i j k l U , ( 2 . 5 4 ) H i k a ( 112 ) r ) i p r s ~ r & J P ~ q ,
and
( 2 . 5 5 ) g i k g j l
- g i j k l ' i l g j k .
The tensors E and H i j can be shown to i j
P r qe q ) U U H 9 'i jpq k l r s
have the following
properties which follow from the symmetries of the Weyl
conformal curvature tensor:
Elementary Causality on Lorentzian Manifolds
A Lorentzian metric g for M is a smooth symmetric tensor
field of type (0,2) on M such that for each p E M, the tensor
g(p) on TpM is a nondegenerate indefinite inner product of
signature (-,+,+,+). It is easily shown that a noncompact
manifold admits a Lorentzian metric. On the other hand for
compact manifolds it can be shown that a Lorentzian metric
exists only if the Euler characteristic vanishes i.e, x(M) = 0.
A spacetime (M,g) is a connected, noncompact, smooth
Hausdorff manifold'of dimension 4 which has a countable basis,
a Lorentzian metric of signature (-,+,+,+) and a time
orientation. We assume noncompactness in the definition of a
spacetime since it can be shown that a compact spacetime admits
closed timelike curves, We assume that M is connected since
disconnected parts of a spacetime should not be able to
interact. A spacetime (M,g) is time-oriented if M has a
nowhere zero, continuous timelike vector field. There is a
large number of inequivalent definitions of spacetime appearing
in the literature. The definition we have adopted is the
weakest one which is either consistent with or can easily be
adapted to the definition used in our major references
[32,33,94,95]. As Hall [I051 points out, there is still some
debate among relativists on the nature of the topology of a
spacetime. The usual manifold topology is a "homogeneous"
topology i . e . for any two points p, q E M there is a
homeomorphism which maps p to q. Thus the usual topology
reflects the locally lR4 nature of M rather than its Lorentz
metric structure which in most cases imposes a local sense of
time-orientation at each point. Gijbel [I061 has studied other
choices of a topology for spacetime which are more compatible
with its Lorentzian metric. As constructed above, the under-
lying manifold of spacetime is a manifold "modelled over IR4".
There are more abstract definitions of a manifold which allow
other "model" spaces to be used [103,107]. Perhaps a "model"
space can be found that intrinsically embraces the notions.
Two spacetimes (M,g) and (M',g8) are equivalent, written
as (M,g) * (M8,g'), if there is a space-orientation and time-
orientation preserving isometry between them. The spacetime
equivalence class of (M,g) is the set
(2.58) [(M,g)I = {(M8,g') : (M'&') (M,ef))*
In general relativity it is the classes [(M,g)] that are
of interest. In practical terms it is not a trivial task to
establish that two spacetimes are equivalent. It is much
easier to establish inequivalence by finding an isometric
invariant that one spacetime has and which the other spacetime
does not have. For a fixed manifold M the set of all
Lorentzian metrics on M is denoted by Lor(M). We write p << q
if there is a future-directed piecewise smooth timelike curve
in M from p to q. We write p S q if p = q or there is a
future-directed piecewise smooth nonspacelike curve in M from p
to q. If p E M then the chronological future of p is the set
(2.59) = {q E M : p << q).
If p E M then the chronological past of p is the set
(2.60) I-(p) = {q E M : q << p).
If p E M then the causal future of p is the set
(2.61) J + ( ~ ) = {q E M : p S q).
If p E M then the causal past of p is the set
(2.62) J-(p) = {q E M : q S p).
A nonzero vector V E T M is called P
(2.63) (a) timelike if g(p)(V,V) < 0;
(2.64) (b) spacelike if g(p)(V,V) > 0;
(2.65) (c) null if g(p)(V,V) = 0;
(2.66) (d) nonspacelike if g(p)(V,V) S 0.
The causal classification of vectors provides a partial
classification of curves in spacetime. If V = a' = a*(a ) then u
we say that the curve a is
(2.67) (a) timelike if g(a(u))(V,V) < 0 for all u E dom(a);
(b) spacelike if g(a(u))(V,V) > 0 for all
u E dom(o);
(2.69) (c) null if g(a(u))(V,V)) = 0 for all u E dom(a);
(2.70) (d) nonspacelike if g(o(u))(V,V) S 0 for all
u E dom(a).
The classification is partial since we do not include curves of
mixed causal nature i.e. a timelike curve continuously joined
to a spacelike curve. The partial classification is sufficient
since we do not observe curves which represent physical
particles changing their causal type.
The elementary causality of a spacetime is defined as the
collection of past and future causal sets and the properties
they induce. A spacetime is chronological if it does not
contain any closed timelike curves through p 6 M so that
p P I+(~). The spacetimes of general relativity are usually
assumed to be chronological on physical grounds. Since compact
spacetimes are not chronological, most of the spacetimes-in
general relativity are non-compact. Note that the term compact
is often abused in general relativity when it is used to refer
to cosmological models whose spatial sections are compact i.e.
Friedmann models.
A spacetime (M,g) is causal if it contains no closed
nonspacelike curves. An open set U c M is called causally
convex if no nonspacelike curve intersects U in a disconnected
set. A spacetime (M,g) with arbitrarily small causally convex
neighbourhoods is called strongly causal. A spacetime (M,g) is
globally hyperbolic if it is strongly causal and J + ( ~ ) n J-(q)
is compact for all p, q E M.
The next notion that we need is the idea of stability
under perturbation of the metric g on M. In order to discuss
this mathematically we need to topologize Lor(M). Since M is
paracompact we may find a fixed countable open covering of M
by coordinate domains 3 = {Ba : a E I) with the property that
only a finite number of the Ba intersect an given compact
subset of M. Thus we have a locally finite subatlas of M. Let
6 : M -----, R+ be a continuous function. Two metrics g and
h E Lor(M) are &close in the fine er-topology if for each
p E M all of the coefficients and all the derivatives up to the
r-th order are 6(p)-close at p when calculated in each of the
coordinate systems (B,,(P,) which contain p. We write
to denote 6-closeness in the fine er-topology on Lor(M), The
sets
(2.71) N (6) {h E Lor(M) : Ig - hir,% 9
< 6)
with g an arbitrary element of Lor(M) and 6 : M ----, R+ an
arbitrary continuous function form a basis for the fine
A spacetime (M,g) is stably causal if there is a fine
eO-neighbourhood N (6) of g in Lor(M) such that each Lorentzian 9
metric h E N (6) is causal. Beem and Ehrlich C331 have shown 9
that a Lorentzian manifold (M,g) with M homeomorphic to fR2 is
stably causal.
The fine er-topologies on Lor(M) for r = 0, 1, 2 may be
interpreted as follows:
0 (a) r = 0 : if h is &close in the fine E-topology then
all the coefficients of h are close to g in the i j i l
fixed covering 3 of M. Thus the lightcones of h and
g are "close".
1 (b) r = 1: if h is 6-close in the fine e -topology then
all the coefficients of hij are close to g and all i j
the partial derivatives hijjk are close to g in ijDk
the fixed covering 3 of M. Thus the Levi-Civita
connections of h and g are "close" hence the systems
of geodesics of h and g are "close".
2 (c) r = 2: if h is &close in the fine &-topology then
all the coefficients of h are close to g and all i j i l
h the partial derivatives hi j , kl are close to
gijjk' gijDkl respectively in the fixed covering 3 of
M. Thus the curvature tensors of h and g are
"close".
There are many more elementary causal properties [ 3 3 , 1 0 8 ]
such as the causal simplicity, causal continuity, future and
past distinguishing properties but the preceding ones are
easily related to a global decomposition of the Lorentzian
metric. The causal properties discussed above are suited to
the Lorentzian warped product construction which we shall
present in the next section. A large class of interesting
spacetimes may be written as a Lorentzian warped product.
The causality relations can be related by a simple
diagram:
Causal
I I Strongly Causal
Stably Causal
Globally Hyperbolic
Figure 7 The Strengths of the Elementary Causality Conditions
Lorentzian Warped Products
Let (M,g) be a Lorentzian manifold of dimension m and
(H,h) be a Riemannian manifold of dimension n. Suppose that
f : M ---- R+ be a smooth function. The Lorentzian warped
product of the first type 1331, M xf H is the manifold
( 3 , s ) = (M x H,g @ fh).
If n : R - M, and r) : fi ----* H are the projections onto
M and N respectively, then we define the metric 2 by
(2.72) I(v,w) = g(n*v,n*ww) + f (X(P) )h(r)*v,r)p),
for v, w E T fi. In a later section we will summarize the P
elementary causality of Lorentzian warped products of the first
type
The Lorentzian warped product of the second type on M x H
is the manifold (fi,g) s (M x H,fg @ h) where f : H ----4 R'. We
define 2 by
(2.73) S(V,W) = f (??(PI )~(~*v,x*w) + h(r)*v,r)*w)
for all v, w E T The elementary causality of warped P
products of the second type has been studied by Kemp [log].
All of the warped products in our work will be of the first
type. As all spacetimes as we have defined them will be
time-oriented we should find a criterion so that a warped
product of the first type is time-oriented, Beem and Ehrlich
1331 have proven the following fact. The warped product M xf H
of (M,g) and (H,h) may be time oriented if and only if either
(a) dim M > 2 and (M,g) is time-oriented; or 2 (b) dim M = 1 and g = -dt .
Let M xf H be a Lorentzian warped product of the first
type. The following is a list (a similar list may be found in
Kemp [log] for warped products of the second type) of useful
properties for warped products of the first type:
(a) For each b E H, the restriction n(r)-'(b) : ~-'(b) --r M is
an isometry of r)-'(b) onto M.
- 1 (b) For each m E M, the restriction r))n-'(m) : n (m) --r H is
a homothetic map of n-'(m) with homothetic factor l/f (m).
(c) If v E T(M x H), then g(np,n*v) 5 z(v,v).
Thus nx : T(M x H) - Twfp)
M maps nonspacelike vectors
to nonspacelike vectors and IE maps nonspacelike curves of
M xf H to nonspacelike curves of M.
(d) For each (m,b) E M x H, the submanifolds n-'(m) and q"(b)
of M xf H are nondegenerate when given their respective
induced metrics.
(e) If 9 : H -----, H is an isometry of H, then the map
cb=i l ( x $ : M x f H - M x f H f H
defined by
Q(m,b) = (m,@(b))
is an isometry of M x H. -f
(f) If p : M - M is an isometry of M such that foq = f then
the map Y = q x \ . M xf H - M xf H defined by is an isometry of M xf H. If X is a Killing vector field
on M (Zxg = 0) with X U ] = 0 then the lift of X to M xf H,
z, such that = (X(n(p),O,,J is a Killing vector
field on M xf H.
( g ) For each b E H, the leaf q-'(b) is a totally geodesic sub-
manifold of M xf H.
Note that in (f) we need fop = f so that f is constant on the
orbits of p otherwise the warping factor is changed.
Elementary Causality of Lorentzian Warped Product Manifolds of
the First Type
Beem and Ehrlich [ 3 3 ] have proven the several propositions
concerning the elementary causality of Lorentzian metrics with
a warped product decomposition of the first type. Let (M,g) be
a spacetime and let (H,h) be a Riemannian manifold. Then
(a) (M xf H, g @ fh) is chronological if and only if (M,g) is
chronological;
(b) (M xf H, g @ fh) is causal if and only if (M,g) is causal;
(c) (M xf H, g @ fh) is strongly causal if and only if (M,g) is
strongly causal;
(d) (M xf H, g @ fh) is stably causal if and only if (M,g) is
stably causal and dim# 2 2;
(e) (M xf H, g @ fh) is globally hyperbolic if and only if
(M,g) is globally hyperbolic and (H,h) is a complete
Riemannian manifold.
There are several more results available in Beem and Ehrlich
1331 . Since the warped products we shall consider in Chapter 4
have Lorentzian factors which are two dimensional, we present
some facts on two-dimensional Lorentzian manifolds. All these
facts are found in [ 3 3 ] .
Let (M,g) be a two dimensional spacetime. Then the
following facts hold:
(a) If M is homeomorphic to R', then (M,g) is stably
causal ;
(b) If M is simply connected, then (M,g) is causal;
(c) If M is simply connected then (M,g) is strongly
causal ;
(d) The universal covering manifold of (M,g) is
homeomorphic to R ~ ;
(e) If M = R ~ , then (M,g) is chronological.
The causal properties of a Lorentzian warped product of
the first type will enable us to examine the elementary causal-
ity of spherically symmetric spacetimes with very little
difficulty provided we can find adequate topological informa-
tion on the factors. This feature of the warped product
representation of metric tensors seems not to be widely
appreciated in the literature. It should be noted that not all
spherically symmetric spacetimes may be globally written as a
Lorentzian warped product. Clarke [I101 has recently produced
an example of a spherically symmetric manifold which cannot be
written as the direct product of two manifolds of lower
dimension. Since the Lorentzian warped product construction
depends on having a direct product decomposition we could not
put a Lorentzian warped product on Clarke's example.
Spherically Symmetric Lorentzian Warped Product Manifolds of
the First Type
A spacetime is spherically symmetric if its isometry group
contains a subgroup isomorphic to the group S0(3), and the
orbits of this group are two-dimensional spheres. The group
action of SO(3) on the orbits may be interpreted as a rotation.
A metric tensor which is invariant under rotations is called
spherically symmetric. A spherically symmetric metric tensor
induces a metric on the orbits which is a multiple of the
metric on a unit sphere. There are several types of coordin-
ates in which it has become customary to represent spherically
symmetric metrics. We will employ the Lorentzian warped
product construction to examine them,
There are two subtypes of Lorentzian warped products of
the first type so that the resulting Lorentzian manifold is
four-dimensional, For dim# = 1 and dimWH = 3 we find the
class of Friedmann-Robertson-Walker-Lemaitre spacetimes if H is
a space of constant curvature. (The naming of these metrics
depends on whether the stress-energy tensor is that of a
perfect fluid or the special case of "dust"). For dimdl = 2
and dim# = 2 we find a class of Lorentzian warped product
spacetimes which includes all of the spherically symmetric
metrics (take H = s') . Let (s2,h) be the standard Riemannian
differential geometry on the sphere. The coordinates we use on
the sphere will be denoted (8,G) so that
(2.74) h = d m 8 + sin28d@d@. Since (M,g) is a two-dimensional Lorentzian manifold we have
four causally different ways of coordinatizing M. If we use
coordinates (t,r) so that i(at,at) < 0 and s(ar,ar) > 0 then we
will call these coordinates TS-coordinates (since at and a are r
timelike and spacelike respectively). In TS-coordinates we can
write the metric on M as -
(2.75) g = -~~(t,r)dt@dt t B2(t,r)drar.
If we use coordinates (t,u) so that
Z(at,at) < 0
and ii(au,au) = 0,
then we will call these coordinates TN-coordinates ( since at
and a are timelike and null respectively). In TN-coordinates U
we can write the metric on M as -
(2.76) p = -~'(t,u)dt@dt + 2B2(t,u)dt@du.
If we use coordinates (u,r) so that
then we will call these coordinates NS-coordinates (since aU
and ar are null and spacelike respectively). In NS-coordinates
we can write the metric on M as -
( 2 . 7 7 ) g = 2~~(u,r)du@dr + B2(u,r)dr&Ir.
If we use coordinates (u,v) so that
Z(au,au) = 0
and &av,av) = 0,
then we will call these coordinates NN-coordinates (since a u
and a are both null). In NN-coordinates we can write the v
metric on M as
(2.78) g = -4~~(u,v)du@dv.
Synge [35] has shown that using NN-coordinates to formulate the
vacuum field equations will directly produce the Kruskal-
Szekeres vacuum solution. As the other coordinate systems have
been extensively studied [111,112] we will confine our
attention to NN-coordinate systems. Thus the metric for the
spacetime which we use can be written in the form -
( 2 . 7 9 ) g = -4f2(u,v)du&iv + ~~(u,v)[dt%de + sin28d+3d+],
thus the metric is a type 1 warped product metric.
From a purely formal point of view there is no difference
between the TN and NS types of coordinates. Goenner and Havas
[I131 point out that the direct integration of the field
equations using a single type of coordinates and the corres-
ponding form of the metric tensor may not produce the optimal
set of solutions. Any spherically symmetric metric can be
written in the NN-coordinates or the TS-coordinates [114]. A
difficulty in working with different canonical forms of the
spherically symmetric metric tensor lies in the fact that one
must prove the inequivalence of the solutions found. It is
well-known in general relativity that this equivalence problem
is notoriously difficult; often more difficult that the process
of finding the solutions themselves. There are various
classification schemes of solutions by which one may prove
solutions inequivalent. When these methods fail there is not
much one can do in practical terms. For the NN-coordinates
there has been little work so this problem is not so pressing
there.
The metric form (2.79) is preserved under coordinate
transformations of the form
Takeno [I141 has shown that the coordinate transformation
(2.80) is the most general that preserves the metric form
Following Beer and Ehrlich [33], we set p = ln(~'), and
1 2 let D and D denote the Levi-Civita connections over the
identity on (M,g) and (H,h) respectively. For vector fields
XI, Y1 E X(M) and X2, Y2 E X(H), we may lift them to vector
fields X = (X1,O) + (0,X ) and Y E (Y1,O) + (0,Y2) in X(M x H). 2 -
Writing D for the Levi-Civita connection of g and using the
Koszul formula (2.30) we find the following formula for D:
(2.81) D,Y = D:~Y, + D~ Y + (1/2) [X,(V)Y~+ Y,(Y)x, x2 2
- Z(x2,y2)gradg~. We use grad tp for the gradient of y on (M,g). We identify the
9 1 vector D y1l, E T M with the vector xi m
E T (M x H). Similar identifications will be (m, b)
assumed.
Decompose tangent vectors X E T (M x H) as X = (X1,X2) and P
define the tensors Hv and hv by
(2.82) Hv(Xl) 3 D1 (gradgV), xi
and
Define the symbol llgradgvll: = g(gradgY,gradgv). Let R, R', and
2 R be the curvature tensors on (M xf H, g @ RZh) , (M,g) , and (H,h). For vector fields X, Y, Z E Z(M x H) we have
(2.84) R(X,Y)Z = DxDyZ - D D Z - D t x , y , Z. Y X
Using (2.83) , the decompositions of vector fields, and (2.84) we find the following formula for the Riemann tensor on the
spherically warped product manifold of type 1:
+ ( 1/4) [yI(v)Z(xz,z2) - X~(~)Z(Y,,Z~) lgradglp.
Similar formulae may be developed for the Ricci tensor [ 3 3 ] and
the Einstein tensor, however these formulae depend explicitly
on the nature of the basis chosen for the tangent spaces
(orthonormal, non-null vectors). For pseudo-orthonormal bases
which include null vectors, one must be careful when writing
the above formulae.
Observers and the Kinematics of their Motion
In this section we formalize the notion of observer and
congruences of observers. We also discuss the well known
Raychaudhuri decomposition of a timelike congruence of
observers. The presentation we follow is a combination of that
of Ellis [39,115], Frankel [98], Szekeres [116], and Greenberg
[lly].
An observer is a time-like curve y : I c R ----+ M such
that g(U,U) = -1, where U = y*(a ) is future-pointing and u is u
the parameter of y called proper time. The vector U determines
a (3+1)-decomposition of TyIu1M = R7 cB R along 7, where we call
R7 the rest space of at y(u). The image y(1) is called the
world line of 7. The timelike unit vector U = y*(a ) is the u
4-velocity of 7. An instantaneous observer is an ordered pair
(p,U) where p E M and U is a future-pointing timelike unit
vector in T M. One can always find a local observer whose P
tangent at p is U.
A reference frame Q on a spacetime (M,g) is a tetrad of
orthonormal vector fields one of which is timelike and whose
integral curves is an observer. A reference frame Q is
geodesic if DTT = 0 where T is the timelike unit
future-pointing vector field in Q. Given an observer 7, we
call a vector field X over y a relative position vector if
x07(u) E RYfu) for all u E dom y and X is invariant under the
flow induced by y. The relative spatial velocity vector along
y is the vector
(2.86) V(X) = FuX
= DUX - g(X,DUU)U,
where U = ~ ~ ( 3 ~ ) .
Let (M,g) be a Lorentzian manifold. For a given open set
'11 s M, we define a congruence in '11 as a family of curves such
that for each point p E 91 there is exactly one curve in the
family which passes through p . Let U be a timelike unit vector
field on (M,g). The acceleration vector of 11 U-observer is the j spacelike vector field A = D,,U i.e. Ai = U .U . For curves in
i ; ~
the congruence of the U-observer the acceleration vector
represents the non-gravitational forces acting on the observer.
Computing the divergence of A we have
By the Ricci identities
and a contraction we find that
We can decompose the "acceleration" tensor Ui with respect to ; j
Uiand h so that C391 i j
where
( 2 . 8 9 ) V a hk h1 U i j i j k;l
is called the relative velocity tensor. The tensor V i .i
represents the relative velocities of particles in the rest
k 3-space of the observer U . We now decompose Vij into symmetric and antisymmetric parts as
follows :
( 2 . 9 0 ) V = eij + oijt i j
where
is called the strain-rate tensor, and where
( 2 . 9 2 ) 0 i j Ulij19
is called the vorticity tensor. Since V is the result of a i j
projection into the rest-space of a U-observer we see that
and
The strain-rate tensor 8 can be further decomposed into its i j
trace and its trace-free parts:
( 2 . 9 5 ) 8 = o + (1/3)8hij, i J i j
where a ij'
the shear-rate tensor, satisfies
and 8 , the volume expansion scalar, is given by
( 2 . 9 7 ) i e = u . ; i
The vorticity tensor oil may be viewed as taking a sphere
of particles in the rest-space of the U-observer into a rotated
sphere of the same volume. A vector oi, called the vorticity
vector, may be associated with oil by the equation
( 2 . 9 8 ) i j k l
i From a knowledge of w we can recover the vorticity tensor by
(2.99) w - ij - 'ijkl ok~'.
The vorticity vector satisfies the following properties:
(2.100) oiui = 0,
and
(2.101) i Oi = 0.
The magnitude of the vorticity is defined by
The shear tensor may be viewed as taking a sphere of
particles in the rest-space of the U-o bserver into an ellipsoid
of the same volume. The direction of any principal axis of the
shear tensor is unchanged but all other directions are changed.
The magnitude o of a is defined by i j
(2.103) oZ (1/2)oiJaij
2 0.
The effect of the volume expansion scalar 9 is to change a
sphere of particles in the rest-space of the U-observer into a
larger sphere so that the logarithmic derivative of the radius
of the sphere is 8 / 3 .
Using the Ricci identity Uiik1 - i l k = j ikl and
multiplying by uk we find
U u k - U u ~ - R ' U u k = O . i;kl i; lk ikl j
Using the definition of the relative velocity tensor V and i j
using the projection hij we find the propagation equation for
V along the integral curve of U: i j
Since (2.105) is a propagation equation for V it also i j
contains the propagation equation of 8, u ij'
and o , We will i j
not find the propagation equation for 61 as we will deal i j
exclusively with spherically symmetric spaces in which the
vorticity tensor o = 0. i j
The propagation equation equation for the expansion scalar 8 is
found from (2.105) by contracting on i and j:
where we have
together with
d8 9 - where dz
used the field equations for a perfect fluid
the definitions above. We have used the notation
T is the proper time along the integral curve of
Equation (2.106) can be written as
(2,107) 2 9' + (1/3)e2 = -20 t A' - (1/2)(p + 3p).
; i
In cosmology it is common practice to define a length scale L
by the equation
(2.108) 8 = 3(lnL)'.
In terms of the length scale L, which we may view as the
distance from our fiducial timelike U-trajectory to a
neighbouring U trajectory (L being measured in R I c r , ) , we have
(2.109) 3La'/L = -20' + A ' - (1/2)(p + 3p). ; i
From this equation we see that shear induces a contraction of
the flow; the divergence of the acceleration shows the tendency
of pressure gradients to cause expansion; the terms from the
trace of the stress-energy tensor show that pressure causes
contraction,
The symmetric trace-free part of (2.105) is the shear
propagation equation. From Ellis [39] we have (neglecting
terms with vorticity)
(2.110) k 1 k
h i hj a i m u * - A f k i l ) ) - A . A 1 j + o i k ~
+ (2/3)Bo + h (-(2/3)02 + (1/3)Ai ) + E = 0. i j ij ; i i j
From this equation we see that the shear is controlled by the
electric part E of the Weyl tensor. Since E represents the i j i j
free gravitational field due to distant matter, these equations
describe the "tidal forces" felt by a congruence of
For spherically symmetric metrics and distributions of
matter we should observe a distortion in the flow of
U-observers as we move toward larger concentrations of matter,
For a spherical fluid element in the rest space of a U-observer
the cross-section of the fluid element orthogonal to the radial
direction should decrease while the radial cross-section should
elongate. The elongation in the radial direction is due to two
effects - the acceleration of the congruence and the tidal forces.
Motions on a Lorentzian Swacetime
The use of symmetries to classify vacuum solutions of the
field equations is well-known in general relativity.
Symmetries may also be used to partially classify interior
solutions, The stress-energy tensor is sometimes assumed to be
invariant under the action of a symmetry i.e. isometric motion.
The problem of how these symmetries effect the individual
matter fields that contribute to the stress-energy tensor is
probably difficult since a given stress-energy tensor may have
several interpretations. We will briefly discuss some of the
symmetries that have been used (a more complete list of
symmetries is in Katzin et al. [29]. In Chapter 5 we will
compute the field equations and other quantities for the case
of nonstatic spherically symmetric anisotropic fluids in
A motion is generated by a Killing vector field X on (M,g)
such that
(2.111) zxgil = 0 .
A conformal motion is generated by a vector field X (conformal
Killing vector field) such that
(2.112) 4gij = 2 W i j 9
where is the scalar conformal factor.
A homothetic motion is generated by an vector field X (homo-
thetic Killing vector field) such that
(2.113) 4 g i j = 2gij*
A special conformal motion is generated by an conformal
vector field X such that
(2.114) xxgij = 2Wij,
y, ij = 0.
where t# is the scalar conformal factor. A conformal collineat-
ion is generated by an affine conformal vector field X such - that
where the collineation tensor H (one could also call H a i j i j
conformal Killing tensor) satisfies
and where I# is the scalar conformal factor. It is still a
difficult open problem to characterize the collineation tensor
Hi Only a few conditions are known which will produce H in i j
such a way as to guarantee the existence of the vector X and
the scalar y. It has proven difficult to find examples of
proper affine conformal vectors (do not reduce of conformal
Killing vectors) until recently when Sharma and Duggal [I181
have provided an abstract example.
An affine conformal vector field is a generalization of a
conformal Killing field. An affine conformal vector field
reduces to a conformal Killing vector field if and only if
Hij z Xgij where X is a constant. We may view the symmetric
tensor Hi, as a measure of how much X fails to be a conformal
Killing vector.
A special conformal collineation is generated by an affine
conformal vector field X such that
(2,119) X i;j
t x j; i = 2Wij + Hij,
where y is the scalar conformal factor and H is the symmetric i j
parallel tensor associated with X and obeys the following
equations
The interest in these more general types of motions is a
consequence of the following theorems. The first is due to
Oliver and Davis [93].
Theorem: Let xi = XU', U'U = -1 and X > 0. A spacetime (M,g) i
admits a timelike conformal motion with symmetry vector xi if
and only if
and a ij'
8, and A i are, respectively, the shear tensor, the
expansion scalar, and the acceleration vector of the timelike
flow generated by Ui.
The second theorem is due to Duggal [91].
Theorem: Let xi = Au', U'U = -1 and X > 0. A spacetime (M,g) i
admits a timelike conformal collineation with symmetry vector
xi if and only if
(a) okl = - (~/3)8~h,,], and (b) = X-'[A t A .u'u. t H ukhji],
D i D J 1 J k
where
= ( ~ e - eX)/3, 8* = (1/2) [Hii t HijUiuJ],
and oij, 8, and Ai are, respectively, the shear tensor, the
expansion scalar, and the acceleration vector of the timelike
flow generated by Ui.
Duggal's theorem is very important for the study of
shearing fluids since it relates the shear of the fluid to a
symmetry i . e . the existence of a timelike affine collineation
vector X and the "affine collineation tensor1' Hij. We conclude
this section with some of the properties of collineations.
An affine conformal vector field is special if and only if
it leaves the curvature tensor R' jkl invariant. A special
affine conformal vector field X is a special case of a Ricci
collineation vector field [29],
(2.123) SeR = O . X ij
The Lie derivative with respect to an affine conformal
vector field X of a non-null unit vector Z is given by C891,
(2.124) XxZ = -(I t ( ~ / ~ ) H ~ ~ Z ~ Z ~ ) Z ' + Y ~ ,
(2.125) qZ i = (I - ( E / ~ ) H ~ ~ Z ' Z ~ ) Z . 1 t H. 1~ . z J t Yi, where Y' is orthogonal to Z ' and s is the indicator of z' .
If v is the flow generated by an affine conformal vector
field X then [I181
(a) a null vector field N will be transformed by v into a
null vector field if and only if H(N,N) = 0;
(b) a non-null vector field V retains is causal character
under q;
(c) two orthogonal vector fields U, and V, will be
transformed into orthogonal vector fields under if
and only if H(U,V) = 0.
There are many more properties of collineations which we shall
omit. These generalized symmetries seem sure to play a very
important role in future studies of realistic fluids. In
Chapter 5 we shall compute the field equations with a conformal
collineation in NN-coordinates for an anisotropic stress-energy
tensor.
CHAPTER III
THE MATHEMATICS OF THE STRESS-ENERGY-MOMENTUM TENSOR -
Classification af the Stress-Energy Tensor
In this section we present a brief summary of the
algebraic classification of the stress-energy-momentum tensor.
The techniques used will be applicable to any second order
symmetric tensor. There are several classification schemes
[27,119,120, 1211 which have been developed in the last twenty
years. The simplest of these is the Segre classification which
uses the eigenvalues and eigenvectors of the Rij with respect
to gij. When working with indefinite metrics one must
prescribe the field over which the Segrb classification takes
place. It is customary to use R, the real field, for the Segre
classification. In general the Segr6 class of R over W i j
will be different from that over C. From the field equations
(1.1) and the definition of the Einstein tensor we see that any
classification of Rij is also a classification of T . In fact i j
an alternative way of writing the field equations is
(3.1) R = T i j
i j - W 2 m i j f
where T I T'~. There is a shift of eigenvalues in passing from
a classification of R to that of T but, as long as it is i j i j
.kept in mind, it poses no obstacle.
Hall [122,124] has proven that in a spacetime (M,g) if
p E M so that Tij 0 at p then there always exists a real null
tetrad {L.,Ni,Xi,Yi} such that T assumes one of the following i j
canonical forms:
where po, pl, p2, p3 E R, and in ( 3.5 ) pl # 0. The first form
(3.2) can be written with respect to a pseudo-orthonormal
tetrad {Ti,Z.,Xi,Yi} where fl~' = L' - N , mi = L' + N' as 1
The forms (3.1) and (3.5) correspond to Segre type
{1;1,1,1) where the numbers inside the braces refer to the
degree of the elementary divisor corresponding to an
eigenvalue. Each elementary divisor corresponds to at least
one eigenvector. If an eigenvalue is algebraically degenerate
(repeated eigenvalue) but the number of eigenvectors equals
the multiplicity of the eigenvalue, then the elementary
divisors for that eigenvalue are of degree 1. The Segre symbol
indicates the algebraic degeneracy by enclosing the degrees of
the elementary divisors corresponding to the degenerate
eigenvalue in parentheses. If the degree of the elementary
divisor exceeds the number of eigenvectors then the degree of
the elementary divisors sum to the multiplicity of the
eigenvalue. In this case at least one of the degrees of the
elementary divisors of the degenerate eigenvalue is greater
than 1. The degree of the elementary divisor corresponding to
a timelike eigenspace is separated from the other degrees by a
semicolon, the others by a comma.
The form (3.3) corresponds to Segre class {2;1,1) which
has a unique null eigenvector in the L' direction. The form
(3.4) corresponds to Segre class (3;l) which has a unique null
i eigenvector in the L direction. In (3.5) complex eigenvalues
occur and x i , and Y' are the only real eigenvectors, thus the
Segre class is written {zS;l,l). For this Tij is
diagonalizable over C but not over R.
We see that T always admits at least two eigenvectors. i j
The Segre class {1;1,1,1) and its algebraic degenerate subcases
is the only class which admits a timelike eigenvector. If
there is no timelike eigenvalue degeneracy (timelike eigenvalue
is distinct from the spacelike eigenvalues) then the timelike
eigenvector is unique.
Energy Conditions
The Einstein field equations (1.1) are usually solved
subject to side conditions. These side conditions will reflect
physical properties of the physical situation we are trying to
model. As we are interested in exact interior solutions, the
"energy conditions" will be applied. There are several types of
distinct energy conditions mentioned in the literature [33,95,
961. Unfortunately not all these energy conditions have
distinct names [96]. The role of all the energy conditions is
to exclude unrealistic models of macroscopic matter.
There is a general consensus among researchers in general
relativity that the energy density of classical macroscopic
i matter as measured by an observer with 4-velocity U is
nonnegative i . e., (3.7) T. uiuJ r O. 1 1
Since we do not admit the existence of privileged observers,
this relation must hold for all timelike vectors ui. These
inequalities are called the timelike weak energy conditions.
Tipler [I231 has shown that the timelike weak energy condi-
tions are the weakest energy conditions that can be locally
defined which use the entire set of timelike vectors in T M. P
If we write the stress-energy tensor T as a i l
where {e } is an orthonormal eigenbasis with e(o, timelike, ( a
then the timelike weak energy conditions for a stress-energy
tensor of Segre class {1,111), and its algebraic degeneracies,
are equivalent [94] to the following system of inequalities on
the eigenvalues X (a) '
(3.9) lo " 0, lo t Xi 2 0, i E {1,2,3}.
The null weak energy condition is
(3.10) r o i for all null vectors K .
The strong energs condition is
(3.11) T .uiuJ t (1/2)T 2 0 i~
for all unit timelike vectors ui. This condition ensures that
the matter stresses will not become so large that R uiuj S 0 , i j
The dominant energy condition assert that T satisfies the the i j
following conditions for each timelike future pointing vector
ui:
(3.12) T uiuJ 2 0, and i j
(3.13) i j T U is nonspacelike. j
This energy condition states that the speed of energy flow of
matter is always less that the speed of light. The dominant
energy condition implies the timelike weak energy condition.
The dominant energy condition excludes stress-energy-momentum
tensors of Segre types {zz;l,l) or {3;1) and their algebraic
degeneracies. It also severely restricts the possible
eigenvalues of Segre classes {1;1,1,1) and {2;1,1) and their
algebraic degeneracies. In particular for Segre class
{1;1,1,1) we must have the eigenvalues satisfying [I241
(3.14) lo L 0, and
(3.15) lXil 5 Xo for i E {1,2,3).
The strong energy condition implies the null weak energy condi-
tions, but does not imply the timelike weak energy condition
* 1961.
A stress-energy-momentum tensor is normal at p € M if
T' .x' is timelike for all nonspacelike vectors X € T M [94]. J P
T is normal if it is normal for every p E M. It can be shown i j
that a normal stress-energy-momentum tensor has a unique unit
future pointing timelike eigenvector [94].
Stress-Energy-Momentum Tensors
The material content of a spacetime is represented by the
stress-energy-momentum tensor T which appears on the right- i j
hand side of the field equations (1.1). T i j
depends on the
fields representing the matter, the covariant derivatives of
these fields, and the metric tensor. Note that T = 0 on an i j
open set U in M means that there are no matter fields on U. By
the field equations we see that the twice contracted Bianchi
identities of the Einstein tensor
(3.16) G ' ~ 0, ; 1
imply that the stress-energy-momentum tensor satisfies a set of
differential identities
It has become common practice in general relativity to
call these identities "the conservation equations" or, for the
case of dust, "the equations of motion". As Wald [96] notes,
the notion of (3.17) as conservation equations is only true in
the differential sense. A better descriptive term for (3.17)
is the "equations of hydrodynamical support".
For a fluid moving through spacetime with a unit-speed
timelike tangent vector U, the flow lines are the integral
curves of the vector field U. We say that the fluid is a
perfect fluid if the stress-energy-momentum tensor has the form
(3.18) Tij = (P+P)U~U~ + Pgij,
where p is the energy density measured by an observer with
velocity U, p is the pressure common to all 2-planes in the
rest-space of the observer.
A viscous fluid with coefficient of dynamic viscosity i
2 0, bulk viscosity E 2 0 , flow vector U , energy density p, isotropic pressure p, shear tensor uij, expansion scalar 0 ,
and heat flow vector Q', has a stress-energy-momentum tensor
given by 11251
(3.19)
where
uiui = -1,
viai = 0 ,
and
i Q1 = 0.
Hall 11191 and Hall and Negm El261 have shown that this
form of the stress-energy-momentum tensor, without any energy
condition imposed, does not restrict the Segr4 class. By using
the projection tensor onto the 3-space orthogonal to ui, i . e .
h I gij + UiUj, we can write the stress-energy-momentum i j
tensor in the form:
Equation (3.20) has the same form as (3.19) if we set
and
with other obvious identifications. From (3.19) it is clear
that setting Qi = 0 forces T to be Segre type {I; 1,1,1) or i J
one of its degeneracies. A stress-energy-momentum tensor for
a viscous fluid without heat flow is easily seen to be normal.
Setting E, q to zero and leaving Qi # 0, we have a
i nonviscous fluid with heat flow. The vector U is not a
timelike eigenvector in this case. Hall and Negm [I261 have
shown that the dominant energy conditions imply that two
physical Segre classes arise from this specialization, namely
2,(11) and {l,l(ll). Only in the last case is the
stress-energy-momentum tensor normal,
Setting the viscosity coefficients 5, q and the heat flow
vector Q' to zero leads to the Segre type (1; (1,1,1)) of a
perfect fluid. It is clear that the stress-energy-momentum
tensor of a perfect fluid is normal.
Hall and Negm [I241 give several combinations of matter
fields and discuss their algebraic structure. The combinations
include two non-zero interacting radiation fields, a perfect
fluid and a radiation field, and two noninteracting perfect
fluids. All of these combinations have stress-energy-momentum
tensors of Segre class { 1 , 1 ( 1 1 ) ) hence are all anisotropic.
For a perfect fluid the timelike weak energy conditions
1961 become
( 3 . 2 1 ) P 2 0,
p + p r o .
For a perfect fluid the strong energy conditions are
( 3 . 2 2 ) P + 3~ 2 0,
p t p r o .
For a perfect fluid the dominant energy conditions are
( 3 . 2 3 ) P 2 !PI 2 0,
For anisotropic stress-energy tensors the situation is more
complicated depending on the composition of the stress-energy
tensor. Hall and Negm [ I 2 4 1 have written the dominant energy
conditions for several cases.
CHAPTER
SHEARING PERFECT FLUID SOLUTIONS IN SPHERICALLY WARPED PRODUCT
MANIFOLDS OF THE FIRST TYPE
The Field Equations in NN-Coordinates - In this chapter we shall write the field equations for a
perfect fluid in NN-coordinates and find some special solutions
which exhibit nonzero shear. From Chapter 2 we know that a
spherically symmetric metric may be written in a variety of
ways when represented by a Lorentzian warped product of the
first type. The advantage of using this representation is that
one can immediately deduce certain aspects of elementary
causality solely from this representation and an analysis of
the causality of the two dimensional Lorentzian factor
manifold.
The metric for the spacetime which we use can be written
in the form -
(4*1) g = -4fz(u,v)du@dv + rz(u,v) [d8@d8 + sinz8d@3d@l.
This metric is a type 1 warped product metric. The coordinates
u and v are null coordinates. There are few papers in the
literature which use double null coordinates [35,1271. The
papers [35,127] have similar calculations which were used to
cross-check the calculations as far as possible.
The metric form (4.1) is not quite the same as that in
[35,127]. In those papers the metric is presented with the
term -4fL(u,v)du@dv replaced by -2f(u,v)duWv. The second
choice allows the possibility that when f changes sign the
coordinates u and v are interchanged.
Beem and Ehrlich [ 3 3 ] have proven the several propositions
concerning the elementary causality of Lorentzian metrics with
a warped product decomposition of the first type. Let (M,g) be
a spacetime and let (H,h) be a Riemannian manifold. Then
(a) (M xf H, g @ fh) is chronological if and only if (M,g) is
chronological;
(b) (M xf H, g @ fh) is causal if and only if (M,g) is causal;
(c) (M xf H, g @ fh) is strongly causal if and only if (M,g) is
strongly causal;
(d) (M xf H, g CB fh) is stably causal if and only if (M,g) is
stably causal and dim# 2 2.
2 For the metric (4.1) we take M to be some open subset of lR
2 with metric g = -4f (u,v)du@dv where (u,v) are the coordinates 2 2
on lR ; for H we take S , the standard sphere with coordinates
($,a), with metric h = d m 6 t sin28d4@d$. With these identi-
fications the all four of the propositions hold when the
appropriate conditions hold on the two-dimensional Lorentzian
manifold (M,g). Recall that sL is complete. The elementary
causality of any perfect fluid solution determined from (4.1)
(as far as the preceding four propositions are concerned) is
determined by simply examining the properties of the (u,v)
coordinates on an appropriate domain, Since the metrics
2 g = -4f (u,v)duMv are all conformal to g = -2duMv on R ~ , the
spaces (M,g) have similar causal properties as 2-dimensional
Minkowski space. If global topological identifications are
made, then great care must be used in trying to use the
preceding theorems. Not all spherically symmetric metrics have
these properties since they cannot all be written as a Lorentz-
ian warped product.
We compute the standard quantities for the metric (4.1)
next. All quantities are computed in the sign conventions of
Chapter 2 (also see Appendix A for notation). The Christoffel
0 1 2 3 symbols of the second kind are (with (x ,x ,x ,x ) = (u,v,8,@))
(4.2) r0 U r O 22 2
00 = 2f /f, = rr /(2f ) ,
v
2 r0 = r O sine, r1 2
33 = sin 8, 33 22 - 2 2 = rr sin 8/(2f ) ,
v = rr sin28/ (2f ') ,
U
r1 2
2 2 = rr /(2f ) ,
u r1
11 = 2fv/f,
r2 02 = ru/r, r2 12 = rv/r,
r2 33
= -cos8sin8, r3 = ru/r, 03
r 13 = rv/r, r3 = cote.
23
Applying the Christoffel symbols we find the geodesic equations
where the prime represents differentiation with respect to an
affine parameter. In the paper of Synge 1 3 5 3 , the Lagrangian
defined from the metric ( 4 . 1 ) is used to discuss radial null
and timelike geodesics for a vacuum in NN-coordinates.
The Killing equations corresponding to the metric ( 4 . 1 )
are
The spherical symmetry implies the existence of at least three
a Killing vectors: E t l ) = W, E ( t )
a a and = sin% + cot8cos~ - a v
a a E ( 3 )
= cos* - cot8sinO - a+* Other Killing vectors may occur
if special assumptions are made on the metric functions f and
r. In the following section we will see that assumptions that
we make in order to render the field equations tractable will
generate an "accidental" Killing vector,
The Riemann tensor has components given by
- R0212 - -rruv,
- 2
R0303 - R0202 sin 8,
R1,12 = 2rr fv/f - rr
v vv ' - 2
R1313 - R1212 sin 8, 2 2 2
R2323 = r sin 8(rurv/f + 1).
We will write the tetrad components of the Riemann tensor
i for a pseudo-orthonormal tetrad { A A 1 chosen as follows:
The independent tetrad-components of the Riemann tensor are
The Ricci tensor components are
The tetrad components of the Ricci tensor are
(4.10) R(oo, = 2r f /(rf3) - rvv v v /(rf2),
R(ol, = -r /(rf2) + f f /f4 - f /f3,
uv u v VU
R ( ~ ~ ) = 2r u f u /(rf3) - ruu/(rf2), 2
R(221 = l/r + r /(rf2) + rurv/(r2f2),
uv 2
R(33, = l/r + r /(rf2) + rurv/(r2f2).
uv
The Ricci scalar (curvature scalar) is
Other invariants constructed from the Riemann tensor are
and the Kretschmann scalar
~~j~~ 4 4 4 2
(4.13) R i j k L = 4(r u r v )2/(r f ) + 8r u r v /(r f )
2 6 2 5 + 16r r f f /(r f ) - 8r r f /(r f ) U V U V U vv u
2 5 2 4 - 8r r f /(r f ) + 4(ruv)2/(r f ) UU v v
2 4 r /(r f ) + 4(f f )2/f8 + 4rUu vv u v
- 8f f f /f7 + 4(f )2/f6 + 4/r4 U V v u v u
= 4[2r f /f-r ][2r f /f-r l/(r2f4) v v v v u u uu
2 2 4 + 4(r )2/(r2f4) + 4[1+r r /f 1 /r uv U v
+ 4[f f -ff 12/fB . u v vu
The components of the Einstein tensor are
(4.14) = 4r f /(rf) - 2ruu/r, u U
2 2 2
Go 1 = 2f /r + 2r r /r + 2r /r,
u v uv
1 = 4r f /(rf) - 2rvv/r,
v v
G22 = -rr /f2 + r2f f /f4 - r2f /f3,
uv U v v u 2
G33 = Gz2sin 8.
We are interested in finding shearing solutions for the
case when the stress-energy tensor is that of a perfect fluid.
Whenever interior solutions are considered one must decide on
the reference frame of the observer. In NN-coordinates we
shall use an observer who is "generalized comoving". This is
a special choice of observer who shares some of the mathematic-
al advantages of comoving observers in TS-coordinates. From a
mathematical point of view, the choice of a comoving observer
does not introduce new unknown functions into whatever problem
is being analyzed. This property is shared by the notion of a
"generalized comoving" observer. A generalized comoving
observer has a velocity vector orthogonal to the orbits of a
symmetry group. For spherically symmetric metrics this means
that the velocity is orthogonal to the two-dimensional orbits
of SO(3). In contrast, a comoving observer is defined to have
velocity vector orthogonal to a hypersurface. In NN-coordin-
ates for the metric (4.1), the velocity vector of a generalized
comoving observer is given by
From now on all the problems to be analyzed will use this
frame. In NN-coordinates an observer who is not comoving in
the generalized sense, but still has velocity orthogonal to the
orbits of S0(3), has velocity given by U i = (a,b,O,O) where
Associated with any velocity vector is a spatial project-
ion tensor hij which completes the (3+1)-decomposition of
spacetime. The components of h for the generalized comoving i j
velocity are
2 2 2
h22 = r ,
h33 = r sin 8 .
Kinematical quantities such as the acceleration vector,
shear tensor, and expansion scalar are computed next. The
nonzero components of the acceleration vector of the
generalized comoving velocity are
(4.17) A, = (fu - f )/(2f), v
*1 = (-f + f )/(2f),
U v
The nonzero components of the shear tensor determined by Ui are
(4.18) (7 0 0
= f(ru + r )/(3r) - (f + fV)/3, v U
(7 0 1
= -f(ru + r )/(3r) + (fu + f )/3, v v
(7 11
= f(ru + rv)/(3r) - (fu + fv)/3,
O2 2 = ( r + r )/(6f) + r2(fu + f )/(6f2),
u v v 2
(7 33
= az2sin 8 .
The expansion scalar 8 is found to be
(4.19) 8 -(ru + r v ( f - f u + f v )/(2f2).
Using the stress-energy tensor for a perfect fluid with
energy density p and pressure p, T ij = (p+p)UiUj + pg,,, p+p # 0, we find the independent field equations to be
The conservation equations are written in the following form:
Notice that the second conservation equation is identically
satisfied if f and r (hence p and p) are functions of u+v.
Solutions of the Field Equations &J-Coordinates
In this section we seek solutions of the field equations
(4.20) to (4.23) which exhibit shear. We rewrite the equations
(4.20) to (4.23) as follows: to find p we add (4.20),
( 4.22 ) , and twice ( 4.21 ) then divide by 4f to get
3 + (r f + r f )/(rf ) - (ruu+ rVV U u v v )/(2rf2)
To find p we add (4.20), (4,22), and subtract twice (4.21) then
divide by 4f2 to get
Taking the difference of (4.20) and
(4.28) fhuu - rVV) = 2r f - U U
Equation (4.28) can also be written
(4.29) 2 (ru/f l U = (rv/f2)vo
Taking the difference of (4.23) and
(4.22) leads to
in the form
(4.27) leads to
(4.30) 2 2 (-f f + ff )/f4 - rurv/(r f ) - (ruu + rvv u v uv / ( 2rf2) 2 + f + r f f 3 - 1 = 0.
v v
Equation (4.30) is the pressure isotropy equation. The
pressure isotropy equation may be written in the form
Using equation (4.29), equation (4.30) can be written as
There are many ways which one may attempt to solve (4.28)
and (4.30) for f and r. Equation (4.29) has a particularly
interesting structure so we shall examine it first. The first
integral of the equation (4.29) is r /fZ = xV, and r /f2 = xu, U v
2 where is a &-function on a contractible domain. If r is a
2 8 -function then we can write (as a consequence of ruv = r ) v u
Defining new variables p = u-v, and t = u+v, we find (4.33)
becomes
We now make the ad hoc hypothesis that f = l?(t). The second of
equations (4.34) can be integrated to find that
Using this in the first of (4.34) and integrating the resulting
linear equation we find
where g and h are arbitrary smooth functions of one variable
and P is given by P(t) ( 1 / 2 ) S L [ g ' (C)/F(S)I~F. Using the
definitions of t and p we have
There is a dual case for the ad hoc hypothesis that f = F(p)
which leads to
(4.38) r(u,v) = g(u-V) - F(u-v)[P(u-V) + h(u+v)].
In both of these cases, use of (4.37) or (4.38) in the
pressure isotropy equation (4.30) leads to an i n t r a c t a b l e
equat ion in general. Thus we have to look for another method
of solving (4.28) and (4.30).
2 For a vacuum one can show that 2f = (1 - 2m/r)U8(u)V'(v)
which leads to Schwarzschild's solution. By a coordinate
change the dependence on functions of u and v can be removed so
that we have f = F(r) = (1 - 2m/r). This motivates us to
assume that for a class of solutions a func t i ona l r e l a t i o n s h i p
e x i s t s between t h e metric c o e f f i c i e n t s f and r even i n t h e c a s e
o f a p e r f e c t f l u i d .
The method that we shall use to study the field equations
is to find a general solution of equation (4.28) under the
hypothesis that a functional relationship f = F(r) exists
between the metric coefficients f and r. Once we have found a
solution to equation (4.28) we shall use it in the pressure
isotropy equation (4.30) to find a mathematical solution to
the field equations. The mathematical solutions will then be
tested to see that appropriate energy conditions are satisfied,
that the shear tensor is nonzero, and that the resulting
solution is nonstatic,
We will define several auxiliary functions which Figure 8
illustrates. For clarity in Figure 8 we have used distinct
symbols for the value of a function and the function itself.
It is a common abuse of notation in applied mathematics to use
the same symbol for both. Thus f = 9(u,v) = FoX(u,v) and
r = R(u,v) as a consequence of assuming that f = F(r). When
convenient we shall employ the usual abuse of notation,
2 Figure 8 Functions related to (ru/f )u = (rv/f2)v.
Let V,(r) = ~'[F(~)l-~dt. Then V,' (r) = [~(r)]-~ > 0 hence 4-
yl exists and is smooth. The value of V, is s = ~(r). We also
define the function.y(s) s $(s) = r. By abuse of notation we
2 2 have sU = r /f and sV = r /f . Thus equation (4.29) leads to U V
(4.39) s - S = o . u u v v
Equation (4.39) has the general solution
2 where g and h are arbitrary & -functions. We find r to be
given by
Using (4.41) we find the following formulae for the partial
derivatives of r and f (we use h' i d(u-v) dh and g' d(u+v) dg )
(4.42) r = f2(g' + h'), u
2 rv
= f (a ' - h'), df 2
r = fZ[(g" + h") + 2f=(ga + h' ) I , u u
df - f 2[ (g" + h") + 2fz(gS - h' ) 21, rvv -
2df f = f =(g' + h'), u
ad f f = f =(g. - h'), v
Putting these into (4.30) we get
adf
Thus, if we find 3 functions f, g, and h so that (4.43) is
satisfied, then we can use (4.41) to define r. This solution
will be a mathematical solution of the field equations only-it
still needs to have the nonstaticity, shear tensor, and energy
conditions checked. Nonstaticity will follow automatically if
g' # 0.
The first case that we shall study is the simple case when
f = 1. By absorbing the constant of integration in (4.41), we
can write
(4.44) r = g(u+v) + h(u-v).
Equation (4.43) reduces to
(4.45) go2 - hJ2 + (g + h)(g.. + h") + 1 = 0.
Differentiating with respect to t = u + v, we find (4.46) g' (g" + h") + (g + h)g"' + 2g'g" = 0.
Differentiating with respect to p = u - v, we find (4.47) hl(g" + h") + (g + h)h"' - 2h'h" = 0.
Differentiating (4.47) with respect to t .gives
(4.48) $' b"' + g"' h' = 0.
If g' # 0 and h' # 0, then h"'/hl = -g"'/g' = -1, where 1 is a
constant of separation. There are three subcases:
(i) X = 0,
(ii ) X > 0,
(iii) X < 0.
For the case X = 0, we have g"' = 0 and h"' = 0 so that we
formally find
Using these solutions in (4.46) and (4.47), we see that
(4.45) becomes
(4.50) 2 2 g' - h 8 + 1 = 0 .
Thus the formal solutions (4.50) have the form
(4.51) g(u+v) = ko + (u+v)sinha,
h(u-v) = c t (u-v)cosha, 0
where a is a real parameter. Using (4.51) we find that r is
given by
(4.52) r(u,v) = co t ko + (u+v)sinha + (u-v)cosha.
Unfortunately, together with f = 1, (4.52) leads to the
unreasonable result that p = 0 and p = 0. Thus case X = 0 does
not lead to a solution.
The case X > 0 leads to the system of equations
(4.53) 2 g"' = a g' , 2 h"' = -a h' , 2 where we have set X = a , a # 0 . This system of equations has
the formal solution
(4.54) U(U+V) -a( u+v g(utv) = ko + kle - k2e 9
2 2 Using (4.54) we can show g" + h" = a (g - h) + a (co - kO). Putting this into (4.46) and (4.47), noting that g"' = aLg'
and g' # 0, we find that g(utv) = (3ko - co)/4 for all values U(u+v ) -Q(u+v)
of u+v. Since the set of functions 11, e 9 e ) is
linearly independent when a # 0 we see that kl = k2 = 0 thus
g = 0 which contradicts the hypothesis g' # 0. Similarly we
can show that h' = 0, contradicting the hypothesis that h' # 0.
Thus there is no solution for the case 1 > 0 when g' # 0 and
h' # 0.
The case X < 0 leads to the system of equations
(4.55) 2 . g"' = -a g ,
h"' = a2h' , 2 where we have set X = -a , a # 0. This system of equations has
a formal solution similar to (4.54) and the same methods may
be used to show that this case has no solution.
If we assume that g' # 0, h' = 0, then we find that
(4.45) reduces to
(4.56) gg" + g. + 1 = 0
where we have absorbed the constant value of h into g.
We set B = g. so that g.' = and (4.56) becomes
(4.57) gpB' + p2 + 1 = 0,
where the prime means differentiation with respect to g . This
equation reduces to the Bernoulli equation
(4.58) B' = -p/g - 8-'/g0 Applying the method in [801 we arrive at the solution
(4.59) - 2 p 2 = C g - 1 ,
. where C > 0 is an integration constant. This equation leads to
(4,60) - 2 g S 2 = c g - 1.
This equation can be completely integrated to give the
2 solution g2 .= C - K2 T 2K(u+v) - (u+v) where K is a constant of integration. The metric has the form
- (4.61) g = -4duMv + [C - K2 F 2K(u+v) - (u+v)~]~wB
Since C > 0, there are two values of u+v for which g would
be zero. Constraining u+v to lie strictly between these values
2 2 guarantees that g > 0. The region g > 0 is an open strip in
2 the (u,v) coordinate plane hence is homeomorphic to R and thus
this solution is stably causal, hence strongly causal, causal,
and chronological.
We now want to check the timelike weak energy conditions.
For f = 1, and r = g(u+v) we find the energy conditions to be
(4.62) p = (1 + g* 2)g-2 2 0,
which will hold for any g # 0, and
'(4.63) ,Y + p = -2g7g r 0. 2
Taking g2 = C - K' T 2K(u+v) - (u+v) and differentiating twice with respect to u+v we find
(4.64) 2
2g' + 2gg" = -2.
Thus
(4.65) 2 gg" = -(1 + g' ) < 0,
and the timelike weak energy conditions hold for the solution
If we apply the dominant energy condition we just have to
check the inequalities
(4.66) ,Y 2 I P ~ 2 0,
which is equivalent to the three inequalities
The first two are satisfied for the timelike weak energy
conditions. All that remains is to check (4.69). We find
that
(4.70) 2 2
p - p = 2/g2 + 2gS2/g + 2gS'/8
= 2(1 t s S 2 + s..)/e2
= 0
from (4.65). Thus the dominant energy conditions hold on the
same region as the timelike weak energy conditions.
If we apply the strong energy condition we must check the
inequalities
Using (4.65) we can show that p + 3p 2 0 on the region where
the solution is defined, thus the strong energy conditions hold
on this region as well.
The energy conditions, (4.67), and (4.70) show that the
stiff equation of state p = p holds. The expansion scalar is
8 = -2ga/g. The acceleration vector A is identically zero.
The shear tensor a is nonzero since a = 2gm/(3g) # 0. By a i j 0 0
transformation to the coordinates t and p it is clear that this
solution is nonstatic since g' # 0.
a a Since f = 1 and r = g(u+v) we see that 5 ( 4 , = - - - au av is
also a Killing vector. E t r ) is orthogonal to the surfaces u t v
constant hence this solution, which is not a dust solution, has
the same symmetry as the Kantowski-Sachs 1271 dust metrics.
The radial (8 = constant, 9 = constant) geodesic equations
can be easily integrated for both the null and timelike cases.
The radial null geodesics are
(4.73) u = b,
V = d,
where b and d are arbitrary constants. The radial timelike
geodesics are given by
(4.74) u(X) = aX + b,
v(X) = cX t d,
where 4ac = 1 and b and d are arbitrary constants.
The scalar invariants are found next. The curvature scalar is
2 2 2 where g = C - K T 2K(u+v) - (u+v) . Other invariants
constructed from the Riemann tensor are
and
= 3 ~ ~ ,
2 2 2 where g = C - K T 2K(u+v) - (utv) , and we have used (4.65) repeatedly.
1/2 For the next example we take f = r . With this
assumption we find that s = lnlrl~ + c, where c is a constant of
integration. Absorbing c into g where s = g(u+v) + h(u-v) we
find from (4.40) that r is given by
where C is a nonzero constant. Using (4.78) in (4.43) we find
after some simplification
(4.79) g " - h'" + g t hU)/2 + e - ( g + h ) /C = 0.
Consecutively differentiating with respect to utv and u-v we
find that either g' = 0 or h' = 0 for all values of u+v and
u-v. This shows that we cannot have both functions g and h to
be nonconstant. If g' = 0 it is easily seen that any solution
which might be derived from (4.79) will be static hence not of
interest in our study. Taking h' = 0, we find the equation
(4.80) g" + 2g. t z ~ - ~ / c = 0.
Setting p(g) = g' and setting U(p) = p L we find the
differential equation
(4.81) dU - + 4U + 4e-g/~ = 0.
Thus pZ = -4e-g/( 3 ~ ) - hence C < 0. There are three
cases: K < 0, K = 0, K > 0.
If K < 0 we can perform a quadrature to implicitly
determine g and hence r. Thus
2 2 where D is an integration constant. Since p = g' 2 0, we see
that K < 0 imposes a restriction on the domain of the solution. 1/2
Using f = r and (4.80) in (4.67), we find that
p 2 0 if 1 + Cg' 'eg 2 0. However the condition p + p 2 0
2 0, hence the timelike cannot be satisfied when 1 + Cg' e
weak energy conditions are not satisfied for the case for any
constant C < 0. Since this argument may be applied for the
cases K = 0, and K < 0, we see that no reasonable solutions 1 / 2 arise from f = r .
1 / 3 1 / 3 Another case is given by f = r . We find that s = 3r
+ c, where c is a constant of integration. Absorbing c into
g(u+v) we find
(4.83) 3
r(u,v) = [g(u+v) + h(u-v)] /27.
Using this in (4.43) and simplifying gives
Let q = h/3, r = g/3, so that f = r + q. Then (4.84) will
reduce to
(4.85) 2 2 + ) + 20") + o r + q)z(2 - q' ) + 1 = 0.
This equation is difficult to solve, so it will be simplified
by putting q .= 0 to get
( 4 . 8 6 ) 2r3r- + ior2r. + 1 = 0 ,
where we use the dot to denote differentiation with respect
da to u+v. Let a = r so that r" = w, where a = a ( T ) . Thus
( 4 . 8 6 ) is reduced to
where we use the prime to denote differentiation with
respect to I'. Dividing by a we find
which we recognize as a Bernoulli equation if we write it as
From [ 8 0 ] we find the solution
where
where K is an integration constant.
Thus we have
so that
( 4 . 9 6 ) a2(r) = cr-lo - 1 / ( 8 r 2 ) ,
where C is a constant of integration. In fact C must be
greater than zero so that a L ( r ) 2 0 . The definition of a shows
that
( 4 . 9 7 ) 2 = Cr-lo - 1 / ( 8 r 2 ) .
Let H(r;C) 3 cr-lo - 1 / ( 8 r 2 ) so that H(r;C) 2 0 . We can
write a solution to ( 4 . 8 6 ) as
Thus f = I' and r = r3 give the solution of the field equations where I' is determined by the quadrature ( 4 . 9 8 ) . The integral
in ( 4 . 9 8 ) is non-elementary when C # 0 .
In terms of I' the mass-energy density p and the pressure p
are given by
( 4 . 9 9 ) p = i /r6 t i 5 r 2 / r 4 ,
and
( 4 . 1 0 0 ) p = - 1 - 1 5 r 2/r4 - 6 r / r 3 .
From ( 4 . 9 9 ) and ( 4 . 1 0 0 ) we see there is no simple equation of
state for this solution.
The timelike weak energy condition p 2 0 is satisfied by
any solution r # 0 of ( 4 . 8 6 ) . The energy condition p + p 2 0
holds if IT'" s 0 . Thus the timelike weak energy conditions
are satisfied for a solution of ( 4 . 8 6 ) if C > 0 and IT" S 0 .
From ( 4 . 8 6 ) we see that r satisfies = - 5 r - 1 / ( 2 r 2 ) < 0
thus the timelike weak energy condition p + p 2 0 holds.
For the dominant energy condition we have just to check
the inequality (since the timelike weak energy conditions hold)
2 0.
Using f = r, r = r3, and (4.86) in (4.101) we find that
(4.101) cannot be satisfied since p - p = -r-= < 0.
For the strong energy condition we just have to check
(4.102) 2 2 2 p + 3p = -2/r - 2 r r f ) - 2 /(rf2)
u v u v
since p + p 2 0 from the timelike weak energy conditions.
Using f = r, r = r3, and (4.86) in (4.102), we find that
(4,103) p + 3~ 7r6 + GOT 2r-4
2 0,
thus the strong energy condition holds.
Any solution of (4.86) with C > 0 will satisfy the
timelike weak energy conditions and the strong energy
conditions. This solution has shear since ooo = (4/3)r',
nonzero expansion since the expansion scalar is 6 = -7T /r2,
and zero acceleration since r is a function of u+v. Writing
the metric in terms of r we have -
(4.104) g = -4r2(u+v)du&iv + r6(u+v) [d6&i6 + sin2t3d~cb].
This metric is clearly nonstatic since r' # 0. As the metric a a
coefficients are functions of u+v we see that t t r , = - is
an additional Killing vector. On each subregion of the (u,v)-
plane where r # 0 the metric (4.104) is stably causal, hence
strongly causal, causal, and chronological.
The Ricei scalar is found, using (4.11) and (4.86), to be
(4.105) R = -301- 2 ~ - 4 - 5r-6. Other invariants constructed from the Riemann tensor are
complicated nonzero expressions of I', r', and I'" which we shall
omit.
a r We next study the case when f = F(r) = e with a # 0.
Equation (4.30) becomes
(4.106) 2ar 2 2 -e /r - rurv/r + arUV + a(r2 u + r:)/r
- (%u + rvv)/(2r) = 0. Simplifying (4.106) we find (assuming that r = R(u+v))
(4.107) 2ar .2 . 2 -e - r + ar2r*. + 2arr - r = 0,
which we rewrite as
We set P = r. so that re. = @ and (4.108) becomes (4.109) r(ar-l)Pr + (2ar-1)p2 - e 2a r = 0,
where the prime means differentiation with respect to r.
Dividing by p and rewriting we get
(4.110) fl' = (1-2ar)P/r(ar-1) + e2av-'/[r(ar-1)]
which is a Bernoulli equation. Equation (4.110) can be written
k = -1. If we set
(4.112) X(r) = (1 - k)[g(r)dr, then the solution of (4.108) is
Thus we are left with the first integral
Evaluating the integral for X(R) we find the following first
integral
.2 - 2 2ar 2 2 (4.115) r = ~r-'(ar-l) t e /(a r ) .
2ar 2 2 Let K(r;a,D) I ~r-'(ar-l)-~ + e /(a r ) . We can write
implicitly a solution r to (4.115) as
(4.116) -1/2 E * (u+v) = S[~(r;a,D)l dr,
where E is a constant of integration. The timelike weak energy
conditions become
(4.117) 2 -2ar 2 -4 p = ~r-4(1+2ar)(ar-l) e + a r (l+ar)'
2 0,
anp, using (4.108) and (4,115)
.2 2 2ar (4.118) p t p = (4arr - 2rra*)/(r e )
2 2 4 3 = 2(ar+l)/(a2r4) t 2D(2a r -l)/[r (ar-1) I
r 0,
These inequalities are very complicated depending on the signs
-1/2 of a, D, r. If we choose D 2 0, then 0 < ar < 2 will
satisfy both (4.117) and (4.118). Thus there is an open strip in
the (u,v)-coordinate plane on which the timelike weak energy
conditions are satisfied. The metric form of this solution is -
(4.119) 2ar(u,v) g = -4e duwv + r2(u,v)[dW9 + sin28d~@],
where r(u,v) is the function implicitly defined by (4.116).
The mass-energy density is given by (4.117). The pressure is
given by
(4.120) 2 -2ar . 2 .2 2 p = -l/r - e [r - 2rr" - 2arr ]/r . 2
Since the open strip is homeomorphic to R we see that the
solution is stably causal hence strongly causal, causal, and
chronological. The shear tensor ail is nonzero since
a = 2earr'/(3r) - 2aearr'/3. The expansion scalar is found 00
-ar to be 8 = -r'emar/r - ae . The metric (4.119) is nonstatic
since both f and r depend on u+v. As the metric coefficients
a a are functions of u+v we see that cf4, = - - - au av is an
additional Killing vector.
The Ricci scalar is given by
(4.121) -2ar .2 R = 2e [r(2+ar)rS* + r + r21/rZ.
Another scalar constructed from the Riemann tensor is
(4.122) -4ar [2ara 2-r'.] 2/r2 + 2e -4ar ..2 2 R ~ ~ R ~ ~ = 2e r [a-l/r]
.. -2ar 2 4 + 2 [ 1 + r " + r r e ] / r .
The scalar R IR jkl is
(4.123) R~~~~ -4ar .2 .. 2 2 2 ..2 R i jkl
= 4 e [(2arr - r r ) + r r
2 4 ..2 2 a r 2 4 + 4a r r + (rm2 + e ) ]/re
From the previous examples we have seen that setting h = 0
(or g = 0) will allow integration of the field equations for
some specific cases. Now we investigate (4.30) more generally
under this hypothesis. Using h = 0 in (4.43), as this will
generate nonstatic solutions, we find the equation
Equation (4.41) yields s = g(u+v) and r = y(s) = y(g). From
dY (4.41) and the Implicit Function Theorem we find a = [F(r)12.
Putting this into (4.124), we get
(4.125)
d where we evaluate - at y(g). If we prescribe F arbitrarily dr
and use (4.41) to find r we see that solving the field
equations reduces to solving the second order autonomous
differential equation (4.125) for g. If we define
(4.126) P " g' = P($) 1
(4.129) C(g) r F(Y(~) 9
(4.125) becomes the Bernoulli equation (for P):
(4.129) d A ~ ~ [ P ( ~ ) I ~ + B(~)[P(P)I~ + c(g) = 0.
We shall analyze (4.129) in two cases: A(g) = 0, and A($) # 0.
dF If we assume that A($) = 0, then F(r) - r- = 0, thus we dr
have F(r) = cr, where c is a constant of integration. Using
this in (4.128) we find B(g) 0, thus (4.129) becomes
(4.130) C(g) = 0.
From (4.130) we must have c = 0 which produces a singular
metric. Thus A(g) a 0 leads to no solutions.
If we assume that A(g) $ 0, and let U(p) = [p(8)12, we
have
This equation is linear an can be integrate to find
- Since U(g) see
that (4.131) imposes some conditions determined by A(g) and
C(g) on the domain of g. Supposing that the domain of g is
nonempty we can find g by a further quadrature. Thus
where D is a constant of integration. We conclude that
assuming that either g or h is identically zero will always
lead to a differential equation of the Bernoulli type
(neglecting the degenerate case when A(g) = 0).
We will apply the above discussion to the study of the
u case f = r , where a is a real number. Equation (4.125) becomes (4.134) (1 - a)rg" + (1 t a - 2a2)r 2u g . 2 t 1 = 0.
I-2a We find s = r /(I-2a) and thus
Putting r into (4.134), yields
where a # 1, 1/2. The exceptional case a = 1/2 has already
been considered and has been shown to lead to no reasonable
solutions. The exceptional case a = 1 has no solutions since
putting f = r in (4.30) directly leads to r 2 = 0. - 1
Now we consider the case when a = -1. Using f = r leads
3 to s = r / 3 + c, where c is a constant of integration. Thus,
absorbing c into the definition of g and letting h n 0, we find
from (4.41) that
Putting this into (4.125) leads to
where we use the dot to denote differentiation with respect
dP to u+v. Setting B = r' so that r" = &, we get (4.139) 2r3@r + 5r2p2 + 1 = 0,
where now the prime represents differentiation with respect to
r. ~ividing by p and rearranging terms we find the Bernoulli
equation
From [80] we find the solution of (4.140) to be
(4.141) p2(r) = yl(r) + y2(r)
where
and l(r) is given by
with C a constant of integration. Thus we find
(4.145) -5r Yl(r) = De ,
(4.146) 5 r
Y2(r) = -e-5rJ[e /r21dr
Thus the'solution of (4.140) is
(4.147) f12(r) = Y,(r) + Y 2 W -5r = De - e-5rJ[e5r/r2]dr.
A first integral of (4.38) is
where D > 0 so that (4.148) makes sense on some subregion of
the (u,v)-plane. Let K(r;D) 3 ~e'~' - e-5rl[e5r/r2]dr so that K(r;D) 2 0. We can write a solution to (4.138) as
This integral cannot be evaluated to find r explicitly in terms
of utv, E, and D, The metric form of this solution in terms of
the implicit function r is
The mass-energy density p is found to be
and the pressure is
so the equation of state p = p/3 holds.
Using (4.138) in (4.67) and (4.68) shows that the
. timelike weak energy conditions hold. The dominant energy
condition holds since
= 2 ( 2r' - rr")
For the strong energy condition we just show that
holds since we have shown that p + p 2 0 for the timelike weak
energy conditions. Thus the strong energy conditions hold on
the region where the solution in defined.
The acceleration vector is zero since r = R(u+v) and hence 2 f = F(u+v). Since aoo = 4ra/(3r ) this solution has a nonzero
shear tensor oil. The expansion scalar is 6 = -r'. The
solution is nonstatic since both f and r depend on u+v. As the
a a metric coefficients are functions of u+v we see 5 = - - -
( 4 ) au av is an additional Killing vector.
The Ricci scalar is
where we have used (4.138).
Another invariant constructed from the Riemann tensor is
2 4 (4.156) RijRiJ = 2[2rS2 + rr'.] + 2r. + 2[l/rZ + rr.' + rm212
r 0.
R'jkl The invariant R i jkl is found to be
(4.157) Rijrl ~ ' j ~ l = 20[2rm2 + rrm.12 + 4r2rmS2 + 4[rra- - r' '1'
r O.
Since the solution is defined only where r' > 0 we see
that the solution is stably causal. From the discussion of
elementary causality in Chapter 2 we see that the solution is
strongly causal, causal, and chronological.
- 2 Now we consider the case when a = -2. Using f = r we
find that
where c is a constant of integration. Solving (4.158) for r,
setting h 0, and absorbing c into the definition of g we find
(4.159) 1/5
r(u,v) = [5g(u+v)l . - 2
Instead of using (4.159) in (4.125), we shall put f = r
directly into (4.30) to get
(4.160) 4 . 2 3r5r" + 3r r + 1 = 0,
where we use the dot to denote differentiation with respect
dB to u+v. Setting fl = r' so that r" = &, we get (4.161) 3r5pv + 3r4f + 1 = 0,
where now the prime represents differentiation with respect to
r. Dividing by f3 and rearranging terms we find the Bernoulli
equation
(4.162) p' = -p/r - 8-'/(3r5). From [80] we find the solution of (4.162) to be
(4.163) pZ(r) = yl(r) + yZ(r)
where
(4.164) A ( r )
Yl(r) = De 9
and X(r) is given by
(4.166)
= -21n(r( + C,
where C is a constant of integration. Thus we find
Thus the solution of (4.162) is
A first integral of (4.160) is
(4.170) - 2 r = ~ r - ~ + (1/3)r-~.
Let K(r;D) s ~ r - ~ + (1/3)re4 so that K(r;D) 2 0. We can write
a solution to (4.160) as
(4.170) C * (u+v) = S[~(r;~)]-"~dr,
where C is an integration constant. This integral can be
evaluated to find r implicitly in terms of u+v, C, and D. The
metric form of this solution in terms o,f the implicit function
The mass-energy density p is found to be
(4.173) 2 p = -3r r + 2r' '),
- and the pressure is
(4.174) 2 - 2 p = r (rr" - 2r ) .
The timelike weak energy inequalities (4.67) and (4.68) become
(using (4.16'0) )
(4.175) 2 .2 l/r223rr L O ,
(4,176) 2 2 8 r - 2r3r.. L 0, These two inequalities imply that
(4,177) 4 2 1 r 3r r' r 0,
and
(4.178) . 2 rr" s -4r S 0.
The inequality (4.177) shows that as r grows in magnitude that
r' tends to zero. The analysis of the timelike weak energy
conditions may be broken into three cases depending on the sign
of the integration constant D. For the case when the
integration constant D = 0, the mathematical solution does not
satisfy the weak energy conditions even for the special subcase
when C = 0. Using D > 0 in (4.170) shows that (4.177) cannot be
satisfied thus D < 0 is the only case left. Using D < 0 in
(4.170) and (4.178) we find that the weak energy conditions will
-1/2 hold on the region defined by 0 < jrl S (-3D) . To check
the dominant energy condition we have to check the inequality
(4,179) 2 2 p - p = 2/r2 + 2r u r v /(r f ) + 2ruv/(rf2)
= 4/(3r2)
2 09
when D < 0 and we have used (4.173) and (4.174)..For the strong
energy conditions it is straightforward to show that (4.173)
2 . 2 and (4.174) imply p + 3p = -6r r < 0 when D < 0.
This solution has nonzero shear tensor a i l on this region
since goo = 2fre3. The acceleration vector is zero since f is
a function of u+v. The expansion scalar 8 = 0.
The Ricci scalar is found to be
(4.180) 2
R = 6 D + 4/r . The scalar R Ri is given by
i J
(4.181) R, j ~ i J = 2[r3r + 4r2r. '1 + Z[r3r.+ - Zr 2 r .2 1 2 + 2[l/r2 + r3r.. + r2r. '1
r 0.
The scalar R, jkl RiJkl is found to be
R~~~~ 4 (4.182) Rijk1
.2 2 = 4r [rr" + 4r 1 + 4r4[3rr" - 2r'2]2 .2 2 6 ..2 + r 4 r - r 1 + 4r r
The final case we shall consider is the case when a = -1/2.
Note that when a = -1/2 that the second term in (4.134) is -1/2 eliminated. Using f = r we find that
(4.183) s = r2/2 + c,
where c is a constant of integration. Solving (4.183) for r,
setting h 31 0, and absorbing c into the definition of g we find
(4.184) r(u,v) = [2g(u+v) l1j2. 2
Since (4.184) implies that r = 2g we must have g > 0. When we
find r we will have to consider the possible branches of this
relation. From (4.134) we get
(4.185) 3rgm'/2 + 1 = 0,
where we use the dot to denote differentiation with respect
2 to u+v. Since r = 2g we have
(4.186) rr' g',
- 2 g" = r + rr",
thus (4.185) becomes
(4.187) 3r2r.. + 3rra2 + 2 = 0,
where we use the branch r+ = m, thus r+ > 0. Note that we
shall only use the subscript "+" on r for clarity. Setting
fl = r' SO that r" = &-, we get
(4.188) 3r2pp' + 3rp2 + 2 = 0,
where now the prime represents differentiation with respect to
2 r. Dividing by 3r fl and rearranging terms we find the Bernoulli
equation
(4.189) p' = -p/r - 2p-'/ ( 3r2).
From [80] we find the solution of (4.189) to be
(4.190) p2(r) = yl(r) + y2(r)
where
(4.191) A ( r ) Y1(r) = De 9
and X(r) is given by
where C is a constant of integration. Thus we find
(4.194) - 2 Yi(r) = Dr ,
(4.195) Y2(r) = -4/(3r).
Thus the solution of (4.189) is
A first integral of (4.187) is
where we must choose D > 0. Let K(r;D) - ~ r - ~ - (4/3r) so that K(r;D) 2 0 defines a region of the (u,v)-plane.. We can write a
solution to (4.187) as
where C is an integration constant. This integral can be
evaluated to find r+ implicitly in terms of u+v, C, and D. The
metric form of this solution in terms of the implicit function
The timelike weak energy inequalities (4.67) and (4.68) become
and
(4.201) p + p = 4/(3r2)
2 0,
using (4.185), (4.186), and the fact that r > 0. The pressure
is
(4.202) p = 1/(3r2)
so we see that the equation of state p = p/3 holds. From
(4.200) and (4.201) we see that the timelike weak energy
conditions hold for the branch r+ = m. For the dominant energy conditions we must check that p - p 2 00
Using (4.187.) we find that
Thus the dominant energy conditions hold for the r = m. Checking the strong energy condition p + 3p 2 0 we find
The Ricci scalar for the branch r = is
The invariant R. .RiJ for the branch r = is 1 J
The invariant Ri jkl Rijkl for the branch r = is
2 (4.207) R i jkl
Rijkl = 4rm2 + 4r"2 + 8(r2r'. + 1/3)
The solution r+ has nonzero shear tensor a on this region iJ
-3/2 since a = r'r 00
. The acceleration vector is zero since f is -1/2
a function of u+v. The expansion scalar 8 = -(3/2)rmr
Since the solution is defined only where rs2 > 0 we see
that the solution is stably causal. From the discussion of
elementary causality in Chapter 2 we see that the solution is
strongly causal, causal, and chronological. The metric (4.199)
is nonstatic since both f and r depend on u+v. As the metric
a a coefficients are functions of u+v we see that 5(,, = - is
an additional Killing vector to those imposed by spherical
symmetry.
2 Now we consider the other branch of r = 2g thus r- = -m and we see that r- < 0. Using r, in (4.134) we get
(4.208) - 3 m g " + 2 = 0.
Using (4.186) we find (4.208) becomes
(4.209) .2 -3r2r*' - 3rr + 2 = 0 .
Setting p(r) = r' so that r" = &--, we get (4.210) -3r2pp' - 3rp2 + 2 = 0 ,
where now the prime represents differentiation with respect to
2 r. Dividing by -3r f? and rearranging terms we find the
Bernoulli equation
(4.211) = -p/r + 2@-'/ ( 3r2) . From [80] we find the solution of (4.211) to be
(4.212) p2(r) = Yl(r) + Y2(r)
where
and X(r) is given by
where C is a constant of integration. Thus we find
(4.217) Y2(r) = 4/(3r).
Thus the solution of (4.211) is
(4.218) ~ ~ ( r ) = y2(r) + y2(r) - 2 = Dr + (4/3r).
A first integral of (4.209) is
(4.219) .2 - 2 r = D r + (4/3r)
where we must choose D > 0 since r < 0.
Let K(r;D) 1 Dre2 - (4/3r) so that K(r;D) 2 0 defines a region
of the (u,v)-plane.. We can write a solution to (4.209) as
(4,220) - 1 / 2 c + (u+v) = J[K(~;D)I dl,
where C is an integration constant. This integral can be
evaluated to find r- implicitly in terms of u+v, C, and D.
The timelike weak energy inequalities (4.67) and (4.68)
become
(4.221) 2 p = l/r
2 0,
and
(4.222) 2 p + p = -4/(3r )
< 0,
using (4.185), (4.186), and the fact that r < 0, Thus the
timelike weak energy conditions do not hold for r- = -m. The dominant energy conditions cannot hold since (4.222) is not
positive. Similarly the strong energy conditions do not hold
for r- = -m, We will no longer consider r- = -me We have looked at several cases of the field equations for
a perfect fluid in NN-coordinates. In each case we see that
the field equations may be reduced to a Bernoulli equation by
prescribing a functional relationship between f and r, If we
make assumptions which force f and r to be functions of u+v, we
are able in some cases to find reasonable solutions of the
field equations which were nonstatic and which have nonzero
shear tensors. In all of the acceptable cases the timelike
weak energy conditions were satisfied. The dominant and strong
energy conditions were applied where possible. In all cases
the shear tensor a was nonzero and the acceleration was is i j
zero. The expansion scalar 8 was computed for all solutions.
For several of the cases some of the scalar invariants such as
R, RijRij and Rijrl R~~~~ derived from the Riemann tensor were
computed.
An interesting observation about all the solutions is that
they all satisfy the condition of being in a T-region [27]
2 since r r i = - r f < 0 when r = R(u+v). This is the , i u v
result of insisting that f and r are functions of u+v so that
the solutions are nonstatic. The known perfect fluid solutions
in a T-region are those of McVittie and Wiltshire 11281 and
Ruban [129,130]. The solution of [I281 has an equation of
state p = (1/3)p. The solution of El291 was a dust solution
p 0, and that of [I301 had a stiff equation of state p = p.
These solutions and the one found here share the property of
having the same symmetries as the Kantowski-Sachs dust
a solutions. The solutions found here corresponding to f = r , a = -2, 1/3 do not have simple equations of state. Ruban has
made several interesting observations about T-regions.
The use of NN-coordinates seems to have some advantage in
the reduction of the field equations. All the calculations
were made with a "generalized comoving" observer. An interest-
ing problem for future work will be to try to use the scheme of
Tupper [18,19] in NN-coordinates to try to find viscous fluid
solutions. A "tilting" observer in NN-coordinates could be
2 taken to be U i = (a,b,O,O) where ab = f .
CHAPTER V_
SPHERICALLY SYMMETRIC ANISOTROPIC FLUIDS NN-COORDINATES
Anisotropic Stress-Energy Tensors Conformal Collineations
In this chapter we shall calculate the field equations for
a spherically symmetric anisotropic fluid in NN-coordinates.
In particular, we shall compute the equations for the case when
the metric tensor admits a timelike conformal collineation
vector parallel to a "generalized comoving" velocity vector.
From Chapter 2 we recall that the equations which
determine a collineation vector X are
where tp is the scalar conformal factor. Hij is the synimetric
covariantly constant tensor associated with X and obeys the
following conditions
If we have the additional condition on the conformal factor
then the collineation vector is "special".
There are different methods in which conformal
collineations may be used to study the problem of finding
interior solutions. The first method is to prescribe the
collineation tensor Hij, assume that the metric admits a
collineation vector x i with collineation tensor Hi j, and then
I
applying this symmetry to the field equations. This is the
method used in [89,90,91,92]. This method assumes that the
matter fields constituting the stress-energy tensor partake in
the symmetry and hence leads to difficult problems of "inherit-
ance of symmetry". A second method is to follow the first
method except for the last step of applying the symmetry to the
field equations. In both of these methods the conformal
collineation tensor is chosen in an ad hoc way. A third method
is suggested by a theorem that we will prove later. Suppose
one were able to construct a collineation tensor H from the i j
stress-energy tensor Tij. Then applying the first or second
methods above, we could seek solutions of the field equations
as before. This third method seems to have the advantage of
eliminating the ad hoc choice of the collineation tensor. It
is clear that the third method will available only for very
special stress-energy tensors since H = 0 is a very iJ;k
restrictive condition on the metric g i j and T . i j
Duggal [91] notes that the existence of a physical
solution of the field equations which admits a proper conformal
collineation vector (not a conformal motion) is a very diffi-
cult problem. Hall [I311 asserts that a proper conformal
collineation exists in a perfect fluid spacetime only for a
stiff equation of state.
The existence of a conformal collineation vector depends
on the existence of a covariantly constant symmetric tensor H i j
other than gij. Katzin et al. [I321 show that if a space
admits a covariantly constant vector field which is the grad-
ient of a nonconstant scalar field a, and F € g3 so that
p' n 0 , then we have
(5.5) X i s P8(a)a , i is a conformal collineation vector with collineation tensor
( 5 . 6 ) H i j % 2F0(a)a ia , , , J
and conformal factor = 0.
It is well-known [27] that reducible spacetimes may admit
a covariantly constant second order symmetric tensor other than
the metric tensor, but the spherical spacetimes we are using
are not reducible. This follows from the fact that the
Lorentzian warped product is not reducible unless the warping -
factor is trivial.
In the remainder of this section we present some facts
concerning the third method of using collineations. In
interior spacetimes one is motivated to look for a relationship
between the stress-energy tensor and the collineation tensor.
For a nonsingular anisotropic stress-energy tensor such a
relationship may exist if the stress-energy tensor is
recurrent (Ti j; = T. .Vk for some vector Vk). 1 J
Walker 11331 and Patterson [I341 have studied the exist-
ence of covariantly constant second order symmetric tensors on
a semi-riemannian manifold. (Eisenhart [I353 studied the case
of riemannian manifolds). Walker [I331 proved the following
Theorem: If (M,g) is a semi-piemannian manifold and if T is i l
(a) symmetric,
(b) not a constant multiple of gij,
(c) nonsingular ( d e t [ ~ ~ 1 # O ) , 1
(d) recurrent,
then H a(Tij - Xg ) is a symmetric covariantly constant i l i .i
second order tensor where a is a scalar and X is one of the
nonzero g-eigenvalues of Tij.
If T is an anisotropic stress-energy tensor we know that i j
T has Segre class {1,1(11)), hence has three distinct eigen- i j
values. Not all Segre class {1,1(11)) tensors satisfy the
condition (c) in Walker's theorem. A radiating dust is of
Segre class {1,1(11)) but has two zero eigenvalues, hence is -
singular. Furthermore, most stress-energy tensors are not
recurrent, hence condition (d) is violated in Walker's theorem
in general. For nonsingular recurrent anisotropic stress-
energy tensors we are led to the following theorem.
Theorem: A nonsingular recurrent anisotropic stress-energy
tensor T in a Lorentzian manifold (M,g) may be used to i l
construct three covariantly constant second order symmetric
tensors of the form
(5.7) HAij = a p i , - A A g ij ) where the XA is one of the three nonzero g-eigenvalues of
Till and aA are certain scalars.
Proof: Following Walker [I331 we define 4 a det[Till/det[g..l. 1 J
Since spacetime is assumed to be four-dimensional we see that
@ is a fourth degree homogeneous polynomial in the components
T i j with coefficients which are functions of the components of
the metric tensor. Since T and gij are nonsingular we have i j
4 # 0. Since T is recurrent, we have Ti j;k = T. .Vk for some i l 1 J
vector Vk. Taking the gradient of O we find that O 4@Vk, ; k
thus V, = ln(01'4) and hence is a gradient. ; k
Define S @-'I4 Tij
. We claim S ij;k
= 0. Taking the
covariant derivative of S we have i j
Thus S is nonsingular, Sij is not proportional to g and i j il'
s ij;k
= 0. Consider now the tensor Hij S - pgij. Each i 9
coefficient zA, A = 1,2,3,4, of the polynomial
is a sum of products of components of S and gij. Since S i j i j
and g are covariantly constant we see that each rA is i l
constant. Thus every root of F(p) ,= 0 is constant and nonzero
since Sij is nonsingular. Now define HAij s S - PAgij, where i j
pA is a root of F ( p ) . It is clear that HAij,, = 0 for each
root pA of F(p ) = 0. T is an anisotropic stress-energy i j
tensor with three distinct nonzero eigenvalues XA SO setting
We are interested in solutions with shear so we should
look for a relationship between the shear tensor and the
existence of a conformal collineation. Duggal [91] has found
such a relation between the collineation tensor of a collinea-
tion vector (parallel to the velocity) and the shear tensor of
the velocity. Duggal's results are summarized in the following
theorem.
Theorem: Let xi = XU', U'U = -1 and X > 0 . A spacetime (M,g) i
admits a timelike conformal collineation with symmetry vector
xi, conformal scalar y, and collineation tensor Hij if and only
(a) okl = (2~)-'[h~~h',~~~ - (2/3)8*hkl] (b) A, = 1-'(1 t X .uJul t Hjkukhji] , i D J
where
and uij, 8 , and Ai are respectively, the shear tensor, the
expansion scalar, and the acceleration vector of the timelike
flow generated by Ui.
From Mason and Maartens [89] we find that condition (a) in
Duggal's theorem is more transparently written as
(a' ) or, = (2~)-'[h',h'~ - (1/3)hi'hk1l~, ,. The condition (a') clearly shows the close relationship between
the shear of ui and the collineation tensor. Loosely speaking,
(a') expresses the relationship that akl is proportional to a
projection of the collineation tensor H . Thus to force a i j
solution of the field equations have have shear it is
sufficient to assume the existence of a proper conformal
collineation parallel to the velocity. Next we consider the
problem of the existence of a timelike collineation vector X
parallel to the velocity U when we are given Hij, the
collineation tensor.
Given a second order symmetric tensor H which is covar- i .i
iantly constant, and a velocity field u', we can determine the
expansion scalar 0 from the kinematics of the congruence of U.
If we seek a timelike collineation vector X = XU we must
determine the scaling factor X. The solution of this problem
is the content of the following theorem.
Theorem: Let (M,g) be a spacetime and suppose that a second -
order symmetric covariantly constant tensor H and a unit i j
timelike vector field U' is given. Then there is a scalar X
so that xi = XUi is an affine collineation vector with
collineation tensor Hi,.
Proof: In 1891 it is shown that under the hypotheses above that
the conformal factor can be expressed as
y = X .u' + (1/2)H u'u', ; 1 i j
and
9 = 3v/X + (1/2X)hij~ . i j
Substituting the expression for tp into the expression for 8 we
have
kg I 3C + (1/2)hij~ i j = 31-.Ui + (3/2)HijUi~' + (1/2)hiJ~ a
J 1 ij
This last expression is a linear partial differential equation
for X: -1 .ui t 8X/3 = (1/2)H. .Uiuj + (1/6)hij~ . Linear ; 1 1 J i j
partial differential equations can, in principle, always be
solved [136,137]. Thus we have shown that given a covariantly
constant symmetric tensor H and a velocity vector field, u', i J
i we can determine a timelike collineation vector X parallel to
the velocity ui with collineation tensor Hij.
The two theorems just proved show that the third method of
using collineations may be useful. A key question which will
temper the application of this method is the characterization
of recurrent stress-energy tensors.
The Anisotropic Field Equations in NN-Coordinates - In this section we shall write the field equations for a
spherically symmetric anisotropic fluid in NN-coordinates. We
shall assume that the velocity of the fluid has a special form
in NN-coordinates is generalized comoving. We shall also
compute the equations for a timelike collineation vector
parallel to the velocity. We will assume that the stress-
energy tensor is that of an anisotropic fluid whose anisotropy
vector S is orthogonal to U and orthogonal to the two-
dimensional pressure isotropy surfaces. Since Ui = (f,f,O,O)
and the metric is spherically symmetric we have Si= (f,-f,O,O).
Thus the stress-energy tensor has the form
(5.8) T ij = (P+q)UiUj t qgij + (p-q)SiSj,
i where SiSi = 1, IJ iu i = -1 and S i U = 0. The eigenvalues of Tij
are X = -p, XI = p, X 0
= q. 2,3
The form of the metric tensor we use is:
which we recognize as a spherically symmetric metric in
NN-coordinates. Since the metric is a Type 1 Lorentzian warped
product, the comments of Chapter 4 on elementary causality still
hold. In these coordinates the anisotropic field equations
become (independent equations only)
(5.10) 2 2 2 p - p = 2r r /(r f ) t 2ruv/(rf2) + 2/r ,
U v
(5.11) 3 2 p t p = 4r f /(rf ) - 2r /(rf ) ,
v v vv
(5.12) 3 2 p t p = 4r f /(rf ) - 2r /(rf ) ,
u U uu
(5.13) q = - ruv
/(rfZ) t f f /f4 - f /f3. u v VU
The conservation equations are
(5.14)
If we assume a timelike collineation vector X = XU with
a collineation tensor H i j , then we find the equations
Note that we have not specified what H is, but are just i j
writing the right-hand side of H = X i; j + X i l j ; i - WgiJ- From the condition H = 0 in the definition of a conformal ij;k
collineation vector we find the following equations
(5.20) Hoo;o = 2fX uu - 2Xf + 8X(f )'/f - 8f X uu u u u Y
(5.21) H 00; 1
=2fX -2Xf - 2 f X + 2 f X uv uv u v v u y
(5.22) Hol;o f X u u + f X vu + Xf uu + X f vu -2X(fu)2/f
- 2Xf f /f + 4f2vu - f X + fvXu, \I v U v
(5.23) 5 fX- uv + fX vv + Xf uv + Xf vv - 2X(fv)2/f -2Xf f /f +4f2Tp + f X - f X
u v v u v v u p
(5.24) H 02; 2
= -rXr f /(2f2) - rXr f /(2f2) + rXrvfu/f2 U u u v
- rr X/(2f) - rr X /(2f) - rr X /f U u u v v U
+ X(r )'/f + Xr r /f, U u v
2 H03; 3 = sin 6H02;2,
H1l;O = 2fX - 2Af t 2f X - 2fVAU,
vu VU u v
H l1;l
= 2fX - 2Xf + 8X(f )'/f - 8f X vv vv v v v *
2
H1Z;2 = rXr f /f - rXr f /(2f2) - rXr f /(2f2)
u v v u v v
- rr X /f - rr X /(2f) - rr X /(2f) u v v U v v
+ X( r ) '/f + Xrurv/f, v
2
H13;3 = sin 6H12;2,
(5.30) 2
H22;0 = rXr f /f t rXrvf /f2 - rXruu/f
U U u
- rXr /f V U
- rr X /f - rrvXu/f u u
- 2r2vu + X(r u )2/f + Xrurv/f,
H 2
22; 1 = rXr u f v /f + rXrvfv/f2 - rXruv/f - rXrvv/f - rruXv/f - rr X /f v v
- 2rZIv t X(r )2/f + Xr r /f. v U v
(5.32) 2
H33;o = sin BH,,;,, (5.33) H
2 = sin BH22,1. 33; 1 ,
These equations, when combined with the field equations,
form a formidable system of partial differential equations.
The equations (5.20) to (5.33) show the severe constraints that
the existence of a conformal collineation place on the
metric. -
The kinematic quantities of the timelike congruence
determined by U are calculated next. The shear tensor of U is
(5.34) 0
= f(ru t rv)/(3r) - (fU + f v
u = -f(ru + r )/(3r) + (fU + f 0 1 v v
u = f(ru t r 3 - f + fV)/39 1 1 v u
2
O22 = -r(r
U + r )/(6f) t r2(fu + f )/(6f ) ,
v V
2 a = sin 8, 33
The acceleration vector of U is
(5.35) = (fu - f )/(2f), v
A1 = (f" - f )/(2f). U
The expansion scalar of the congruence determined by U is
It is customary to impose some energy conditions on an
interior solution in order to give some physically plausible
properties to the solution. As pointed out in Chapter 3, there
are several alternatives available. We shall write the time-
like weak energy conditions as they are the most straight-
forward to apply. The timelike weak energy conditions for the
field equations (5.10) to (5.13) are
(5.37) 2 2 3 0 s p = r r /(r f ) + (rufu + r f )/(rf )
u v v v
+ ruv /(rf2) + l/r2 - (r uu + rvv)/(2rf2), 3
0 i p + p = 2r f /(rf3) + 2rvfv/(rf ) - r /(rf2) u u U U
- rvv/(rf2), 2 2 3 0 s p + q = r r /(r f ) + rufu/(rf ) - r /(2rfz)
U v U u
+ l/r2+ r f /(rf3) - rvv/(2rf2) + f f /f4 v v U v -
- f /f3. vu
The dominant energy conditions are given by the five
The strong energy conditions are given by the inequalities
(5.41) P + P r O ,
P + S r : O ,
p + 2 p + q r o .
A Study of the Conformal Collineation Equations
In the first section we defined a conformal collineation
by equations (5.1) and wrote the condition H = 0 in i j ; k
NN-coordinates in equations (5.20) to (5.33). In this section
we will study the system (5.20) to (5.33) with a view to
finding nonstatic shearing anisotropic solutions of the field
equations (5.10) to (5.13) which admit a conformal collineation
vector parallel to the generalized comoving velocity. Our
approach is to study (5.20) to (5.33) and find candidate cases
with nonzero shear tensor. These candidate cases will be used
in the field equations and appropriate energy conditions.
If we subtract (5.26) from (5.21) we find that
-
This equation can have solutions in only five possible ways
depending on the number of partial derivatives of f-and X which
are zero. We will use (5.42) to systematically attempt to
solve (5.20) to (5.33). With the understanding that c and k
represent nonzero real numbers we define several cases for
solving (5.42) as follows:
case A: f = c # 0, X = k # 0, c,k E W;
case B1: f = c # 0, and X = 0, X # 0; u v
case B2: f = c # 0, and X # 0, X = 0; u v
case C1: X = k # 0, f = 0, f # 0; U v
case C2: X = k # 0, f # 0, f = 0; u v
case Dl: X = k # 0, f # 0 , f # 0; u v
case D2: f = c # 0, XU # 0, 'h # 0; V
case D3: fV = X = 0 , fU # 0, X # 0; v u
case D4: fU = X = 0, f # 0, X # 0; u v v
case El: f # 0, X # 0, f # 0, X # 0, fA - A' = 0; U u v v
case E2: f # 0, Xu # 0, f # 0, X # 0, fA - A' # 0. U v v
We will consider Case A first. As a consequence of the
hypothesis for case A, (5.20), (5.21), (5.26), and (5.27) are
satisfied. Equations (5.22) and (5.23) imply that is a
constant. Equations (5.24) and (5.28)imply that r satisfies
the following system of equations:
At this point case A can be broken into four subcases depending -
on r:
subcase (1): ru = 0, rV = 0;
subcase (2): ru # 0, r = 0; v
subcase (3): ru = 0, r # 0 ; v
subcase (4): r 0 r $ 0 , r + rv = 0. u v u
For subcage (I), r is a constant. From (5.34) we see that
a = 0, thus this subcase is discarded. i S
For subcase (2), equation (5.43) produces ru = 0 which
contradicts the hypothesis of subcase (2). A similar conclusion
holds for subcase (3),
The hypothesis of subcase (4), together with (5.43), implies
that r = p(u-v). Equations (5.30) and (5.31) are satisfied by
r = p(u-v), but (5.34) shows that we have a i j = 0, hence we
discard this subcase.
The preceding four subcases show that case A does not
admit any solutions which lead to nonstatic shearing solutions
of the field equations.
Next we consider Case B1. Since 1 = 0 and XV # 0 we see u
that X = A(v). Equations (5.20), (5.21), and (5.26) are
satisfied. Equations (5.22) and (5.23) imply
(5.44) ~ ( v ) = -A'/(4f).
Equation (5.27) shows that A(v) = av + b, a# 0. Equation
(5.24) reduces to
(5.45) r [-a/(av + b) + 2(r + r )/r] = 0. U U v
Equation (5.28) becomes -
(5.46) (av + b)r I-a/(av + b) + 2(r + r )/r] = 2aru. v u v
Now case B1 can be broken into four subcases depending on r:
subcase (1): r = 0, rV = 0; U
subcase (2): ru # 0, r = 0; v
subcase (3): ru = 0 , rv # 0;
subcase (4): ru # 0, rV # 0.
For subcase (I), r is a constant (which we may assume is
nonzero) and thus (5.45) and (5.46) are satisfied. Equation
(5.31) implies that % = 0 thus -AJ/(4f) = 0 and a = 0 which
contradicts the hypothesis Xv # 0 of case B1. Thus subcase (1)
is discarded.
For subcase (2), using ru # 0 and r = 0 in (5.45) and v
(5.46), immediately leads to a contradiction hence this case is
discarded.
For subcase (3), equation (5.45) is satisfied and (5.46)
has the solution
where K # 0. Equation (5.31) is reduced to 2a/(av t b) + 1 = 0
which cannot hold for arbitrary values of v. Thus subcase (3)
is discarded.
For subcase (4), equation (5.45) reduces to
If (5.48) holds, then from (5.46), 2aru = 0 which either
contradicts a # 0 or the hypothesis that rU # 0. Thus we
discard subcase (4). The case B1 does not lead to any
solutions.
Case B2 is entirely dual to case B1 so we conclude that
case B2 leads to no solutions.
We now consider case C1. Since f = 0, f is a function of U
v alone. By a coordinate transformation we can reduce this
case to the case when f is a constant. Since X is a constant,
we see that case C1 reduces to the case A which has no
acceptable solutions. We thus discard case C1. As case C2
is dual to C1 we discard it as well.
Now we consider case Dl. Equations (5.20) implies that
for some function G. Similarly (5.27) implies that
(5.50) (f-3)v = H(u),
for some function H. The integrability of (5.49) and (5.50)
implies that
where K i s aconstant, Thus G(v) = Kv + a, H(u) = Ku + b so
(5.51) may be integrated to find
(5.52) -113
f(u,v) = EKuv + au + bv + c] 9
where a, b, c are constants of integration. Equation (5.21)
implies that fUv = 0. Using (5.52) we find K = 0 and ab = 0.
If a = 0, then f = 0 contradicts the hypothesis that fU # 0. u
A similar result holds for b = 0. Thus case Dl is discarded,
For case D2 we have (5.20), (5.21), (5.26), and (5.27)
leading to
2 where a, b, c are constants of integration and a2 + b > 0.
Equations (5.22) and (5.23) imply that is a constant.
Equation (5.24) reduces to
Equation (5.28) reduces to
At this point case D2 can be broken into four subcases depending
subcase (2): rU # 0, r = 0; V
subcase (3): rU = 0, r # 0; v
subcase (4): ru # 0, rv # 0.
For subcase (I), r is a constant (which we take to be
nonzero). Equations (5.30) and (5.31) imply that q is a
constant. Equation (5.34) implies that the shear tensor a is i j
zero so we discard this subcase.
For subcase (2), equation (5.55) implies b = 0 so that
(5.53) gives A(u,v) = au + c. Equation (5.54) can be
integrated to find
(5.56) 1 /2
r(u,v) = A(au + C) ,
where A # 0. Equation (5.31) is satisfied and (5.30) can be
integrated to find
(5.57) k
r(u,v) = B(au + C) , where B is a nonzero constant of integration. Comparing (5.56)
and (5.57) we see that if A = B and k = 1/2 then we have a
solution of (5.20) to (5.33). However we have seen that
X(u,v) = au + c, thus A = 0, which contradicts the hypothesis v
of case D2. Since subcase (3) is dual to subcase (2), we will
not discuss it further.
For subcase ( 4 ) , equation (5.30) implies
(5.58) (ru + rvIu + dr,, + rv)/l - rub,, + rv)/r = 0.
Similarly equation (5.31) implies
(5.59) (ru + rv)v + b(r + r )/A - rv(ru + rv)/r = 0. u v
Consider the case when rU + rv = 0. Then (5,58) and (5.59) are
satisfied. Equation (5.54) becomes
(5.60) -ru(a + b)/2 = ar v '
and equation (5.55) becomes
(5.61) -r (a + b)/2 = br . v U
We can solve this linear system of equations for rU and rV only
if a = b. This leads to r(u,v) = p(u-v). From (5.34), we find
the shear tensor o is zero since f is a constant. i 3
Now consider the case when rU + rV # 0 . Equation
implies
(5.62) ('U + r ) /(rU + r ) + a/l - ru/r = 0, v U v
which may be rewritten as
(5.63) [lnlru + rVllU = [lnlr/~I~ u . Similarly equation (5.59) reduces to
(5.64) [lnlr,, + rVllV = [lnlr/~l~,,.
The integrability of (5.63) and (5.64) is clear so we obtain
where K # 0 is a constant of integration.
Using (5.65) in (5.54) and (5.55) gives
(5.66) ru[-(a + b)/2 + K] = arv.
and
(5.67) r [-(a + b)/2 + K] = br . v U
Equations (5.66) and (5.67) are linear in rU and rV and can be
solved for ru and rv only if
(5.68) K = (a + b f m ) / 2 .
Using K in (5.65) leads to
(5.69) r(u,v) = p(au + w) ,
where a K - (a + b)/2. Thus (5.69) leads to a solution of
(5.20) to (5.33) which gives nonzero shear in general.
All that remains is to check the field equations and energy
conditions.
Now we consider case D3. Since fU = 0 we see that f is a
function of v alone. By a coordinate transformation we can
reduce this case to the case when f is a constant. Since X is
a constant we see that case D3 reduces to the case A which has
no acceptable solutions. We thus discard case D3. Since case
D4 is dual to D3 we discard it as well.
Consider case El. From (5.42) that f X - f X = 0, and U v v u
f # 0, f # 0, X # 0, and X # 0. The solution of (5.42) U v U v
under these conditions is X = A(f). Equations (5.21) and (5.26)
imply that
(5.70) fX - Xf = 0. uv uv
Using X.= A(f) in (5.70) yields
(5.71) (fA' - A)fuv + A"ff f = 0, U v
where the prime represents differentiation with respect to f.
The case El is characterized by
(5.72) fA' - A = 0.
Solving (5.72) we find that A(f) = af, a # 0. Thus A" e 0 so
equations (5.21) and (5.26) are satisfied. Equations (5.20)
and (5.27) are also satisfied. Equation (5.22) implies
(5.73) Vu = -(a/2)r(f u + f v ) ~ f ~ u ,
and equation (5.23) produces
(5.74) % = -(a/Z)[(f u + f V ) / f ~ v o
The integrability of (5.73) and (5.74) is satisfied and we find
(5.75) Y(U,V) = -(a/2)C(fu + fv)/fl + c,
where C is a constant of integration. Equation (5.24) reduces
Equation (5.28) reduces to
(5.77) rv[-(fu + f v )/f + (rU + r v )/rl = 0.
Equation (5.30) becomes
and equation (5.31) becomes
(5.79) y~ = -(a/2)C(rU + r v )/rlV.
The integrability of (5.78) and (5.79) is satisfied and the
solution is
(5.80) Y(U,V) = -(a/2)C(ru + rv)/rl + K
where K is a constant of integration. Equations (5.75) and
(5.80) imply ,
At this point case El can be broken into four subcases depending
subcase (2): ru # 0, r = 0; v
subcase (3): ru = 0, rV # 0;
subcase (4): ru # 0, rV # 0.
For subcase (I), r is a nonzero constant, and equations
(5.24) and (5.28) are satisfied. Equation (5.81) implies
(5.82) (C-K)(u+v)/a f(u,v) = G(u-v)e f
where G is an arbitrary c2-function. From (5.34) we find that
Ooo = -(fU + fV)/3 # 0 in general. We still have to check the
field equations and energy inequalities for this subcase.
For subcase (2), equation (5.76) implies that K = C and
(5.83) ru/r - (fU + f v )/f = 0.
Since r = p(u) we see (5.83) has the solution
(5.84) f(u,v) = H(u-v)P(u), 2 where H is an arbitrary f3 -function. From (5.80) we see that
From (5.34) we have Goo = 0. Thus we discard this case.
Subcase (3) is dual to the subcase (2) so we shall
consider it only in the sense that arbitrary functions of u are -
replaced with arbitrary functions of v etc.
For subcase (4), equations (5.76) and (5. -77) imply that
K = C. From (5.76), (5.77), and (5.34) we see that oil = 0 /'
hence we discard this case.
Finally we consider case E2. The condition fh' - A # 0
lead to three subcases:
subcase (1): A" = 0;
subcase (2): A" # 0, f2h" + 4(A - h'f) = 0;
subcase (3): A" # 0, fLh" + 4(A - h'f) # 0.
For subcase (1) we find A(f) = af + b, where both a and b are nonzero. Equations (5.21) and (5.26) imply fuV = 0 which
has the solution
Equation (5.20) and (5.86) lead to
where c, and dl are constants of integration.
Similarly, (5.27) and (5.86) lead to
-1/3 (5.88) h(v) = (C,V + d2) 8
where c2 and d2 are constants of integration. Now we define a
coordinate transformation by
(5.89) U = clu + dl ?
V = C V + d2, 2
and redefine f and r as functions of U and V. From (5.87) and
(5.88) we have
(5.90) f (U,V) = k[u-'j3 + v-lI3] , -
where k2 = c,c2 > 0 . From equation (5.22) we find
3 (5.91) Vu = -(a/2)[fu/fl, - (b/4)[fu/f21u + (af + b)fufv/f a
From equation (5.23) we find
3 (5.92) % = -(a/2)[fv/fly - (b/4)[fv/f21v + (af + b)fufv/f . The integrability of the (5.91) and (5.92) cannot be satisfied
by a function of the form (5.90), Thus this subcase is
discarded.
Subcase (2) assumes A" # 0 and the equation
(5 93 f2h" + 4(A - A'f) = 0
which has the solution
4 (5.94) A(f) = af + bf , where a # 0 and b # 0 are constants of integration. Equations
(5.20) and (5.27) imply that
(5.95) f(u,v) = cuv + du + klv + k2,
where c, d, kl, k2 are constants of integration. Equations
(5.21) and (5.26) imply
Using (5.95) in (5.96), equation (5.96) has no solutions. Thus
we discard this case.
Finally for subcase (3) we define a function X so that
(5.97) x"/x' fA1'/(fA' - A ) - 4/f. Equation (5.97) has a solution given by
Viewing x as a function of u and v we see that by abuse of notation, and (5.20), we may write ~ ( u , v ) = A(v)u + B(v). -
Equation (5.27) implies
At this point we consider two subcases:
subcase (1): AU(v) f 0;
subcase (2): A"(v) = 0.
For subcase (1) we see u = -BU(v)/A"(v) which is
impossible, so we discard this subcase.
For subcase (2) we have A(v) = cv + d, and B(v) = klv + k2.
Since X is locally invertible we may write
(5.100) f (u,v) = xC(~(v)u. + B(v))
= xC(cuv + du + klv + kz).
If c = 0 we can show using (5.21), (5.26), and (5.20) that
fh' - A = 0 which contradicts hypothesis. When c # 0, we
define a new variable
(5.101) 5 3 cuv + du + kjv + k2.
Using (5.20), (5.21), and (5.26), we can show by a complicated
argument that
(5.102) f(u,v) = KF-"',
where K is a constant. From (5.102), f may be written as
where a, B , 7, and 6 are constants, Defining new variables
(5103) reduces to the case when f is a constant, which is a
case already discussed.
Now we shall examine the field equations and the energy
conditions for the few cases above which might admit a conformal -
collineation vector parallel to the comoving velocity, We
shall rewrite the field equations in a more convenient fashion.
Subtracting (5.11) from (5.12) we find after simplification
( 5.10d~) f(ruu - r~~ ) = 2 r f - 2 r f . U U v v
Equations (5.10), (5.11), and (5.12) determine p,
(5.11) plus (5.12) minus twice (5.10) determines p,
and (5.13) determines q,
(5.107) 3
q = - r /(rf2) + f f /f4 - fvu/f , uv u v
conditions.
We return to subcase (4) of case D2, we have
where a n K - (a + b)/2. Using (5.106), and the fact that f is
constant, in (5.104), we find that a = fat. BY a coordinate
transformation we can take a = 1. If a = -1, (5.104) is
satisfied, but the solution is static and the shear tensor is
zero. If a = 1, equation (5.104) is satisfied and the shear
tensor a will be nonzero. The solution is nonstatic since r i j
is a function of u+v. The energy conditions will be examined
later.
For subcase (1) of case El, we have r is a nonzero
(C-K)(u+v)/a constant, and from (5.82), f(u,v) = G(u-v)e , where
2 G is an arbitrary &-function. Equation (5.104) is satisfied.
and the shear tensor a is nonzero. The energy conditions i j
will be examined later.
For subcase (2) of case El, r = p(u) and from equation
(5.84), f(u,v) = H(u-v)p(u), where H is an arbitrary 2
& -function. Using r and f in (5.104), we find
where the prime denotes differentiation with respect to u-v,
and the dot denotes differentiation with respect to u.
Equation (5.108) is separated in terms of the variables p u-v
and u. Equation (5.108) may be integrated to find r and f such
that a transformation of variables leaves r = constant and f as
a function of p. Any such solutions will be static, so. we
discard this case. Since subcase (3) of case El is dual to
subcase (2) of case El, we also discard it.
The only cases which remain to apply the energy conditions
to are:
(a) subcase (4) of case D2 with a = a = 1 ;
(b) subcase (1) of case El.
Consider subcase (4) of case D2 with a = a 1. We have f
is a constant and from (5.69), r(u,v) = p(u + v). We shall
take f = 1 (we can always do this by a coordinate
transformation). The timelike weak energy conditions with f = 1
and r(u,v) = p(u+v) in (5.37), (5.38), and (5.39) -
produces
We see that any function which satisfies pp" S 0 will
satisfy the timelike weak energy conditions. For the dominant
energy conditions we just have to satisfy the inequalities
(5.112) 2 2 p - p = 2r r /(r f ) + 2ruv/(rf2) + 2/i2
u v
2 0,
Using f = 1 and r(u,v) = p(u+v) in (5.112) and (5.113) yields
(5.114) p - p = 2(1 + p%$ + 2pm/p
The dominant energy conditions will hold if
For the strong energy conditions we just have to check the
inequality
Using f = 1 and r(u,v) = p(u+v) in (5.117) yields
The strong energy condition will hold if
The form of the metric for the solutions defined by the energy
inequalities is
This metric i s nonstatic and has an extra Killing vector given
a a = - - - by 5 4 , avo The Ricci scalar is given by
The scalar R R" is given by i f
The Kretschmann scalar is
The nonzero components of the shear tensor a are i j
(5.124) a 00 = 2p8/(3p),
a o 1 = -2~'/(3~),
O t I = 2p1/(3p), -
0 2 2 = -PP'/~,
a 2
33 = 0 sin 8 .
22
The acceleration is identically zero. The expansion scalar is
e = -2pf/p.
Now we consider subcase (1) o'f case El. For this case we
may take r = 1 as the simplest case of a constant r, and (C-K)(u+v)/a f(u,v) = G(u-v)e , where G is an arbitrary
2 ?%-function and a # 0. If we define a constant
u(u+v) a = (C - K)/a, then we have f(u,v) = G(u-v)e
The field equation (5.102) is identically satisfied by r = 1 Q(u+v) and f(u,v) = G(u-v)e . The timelike weak energy
conditions become
The pressures are
The dominant energy condition give
The class of functions G which satisfy dominant energy
a(u-v) conditions is nonempty since G(u-v) = e is in it. The
Ricci scalar is
-2a(u+v) (5.132) R = 211 t e (GG" - G' ')/G~]
- = 2 ( ~ + 9).
Other invariants constructed from the Riemann tensor are
(5.133) -4a(u+v)
R ~ ~ R ~ ~ = 2e (GO" - GJ 2, 2 / ~ 8
2 = 2q , and
~~j~~ = 4 + 4e -4a(u+~) (5.134) R i ~ ~ ~ (GG" - G' 2)2/~8 = 4(l t q2).
The nonzero components of the shear tensor oil are
(5.135) - - - 2 a G e ~ ( ~ + ~ ) /3, Oo 0
a(u+v) u 0 1
= 2aGe / 3 ,
a( u+v) u = -2aGe 11 /3,
O 2
33 = o sin 8.
22
The nonzero components of the acceleration are
The expansion scalar is
The two cases, which lead to mathematically reasonable
solutions of the field equations and which admit a timelike
collineation vector parallel to the velocity, produce reducible
spaces. This result is in agreement with recent unpublished
(as yet) results of Hall and da Costa 11441 who have shown that
a collineation tensor H with H i j;k
= 0 exists on a spacetime i j
if and only if it is reducible. The result of Hall and da -
Costa shows that the usefulness of collineations will be fairly
limited.
\, The recent advance made by Duggal in relating the
existence of a timelike conformal collineation vector to the
shear of the fluid velocity may be useful in future studies of
shearing solutions in reducible spaces. The ability to make a
mathematical hypothesis, about the symmetries to be imposed
upon a problem, which leads to a nonzero shear tensor, should
eliminate many ad hoc hypotheses which have been used to find
solutions with shear.
Since the propagation of the shear tensor is closely
connected with the electric part of the Weyl tensor Eij,
Duggal's theorem shows that when a timelike collineation vector
parallel to the velocity exists, the electric part of the Weyl
tensor E i j is partially determined by H i , . Penrose [I381 has
also conjectured that there is a relation between gravitational
entropy and the Weyl tensor. Since the magnitude of the Weyl
tensor is a rough measure of the "clumping of matter", further
studies of shearing solutions may lead to results related to
the conjectures mentioned in Chapter 1.
APPENDIX A_
TRANSFORMING SYSTEMS OF NOTATION IN GENERAL RELATIVITY
A frequently frustrating problem in general relativity is
the nonstandardization of notation. There are hundreds of
different notation systems available depending on the selection
of a few parameters. It is a common piece of folklore in
general relativity that it is "just a matter of fiddling to get
the signs right". Any survey of the literature will reveal
that many authors do not supply adequate information to
decipher their results. In recent years, the influential text
11251 has had a remarkable standardizing effect on this problem.
The use of computers to perform calculations of evermore
increasing complexity has also stimulated standardization. The
system we have created here is not the most extensive possible
- it would take 4 or 5 more "switches" to even approach a reasonable degree of completeness. The system presented in the
following is a minimal system that will work for all the
"standard calculations".
In this appendix the work of Ernst 11391 and Misner et
al.[125] is generalized and unified in a manner similar to that
in Biech [140]. The advantage of this unified "translation
formalism" is that computer programs can be written which are
"independent" of notation conventions. In [I401 it was noted that
almost all systems of notation in current use in general
relativity can be classified by six parameters el, i = I,.. .,6.
These parameters take the values 1 or -1 depending on the
notation convention. The five parameters ci, i=1,2,3,5,6 are
independent and deal only with mathematical definitions. The
parameter c4 is used as an auxiliary parameter in order to
unify this classification with that of Ernst C1391. The three
parameters defined in Misner et al. [I251 are denoted by Wi,
i = 1,2,3.
The sets of parameters {cl, E ~ ~ E ~ ~ E ~ ) and {W1, W2,W31 are
sufficient to transform any tensor expression which does not
involve the Levi-Civita tensor density. Transformations of
tensor expressions involving the Levi-Civita tensor density use
the parameters c5, E ~ . The definitions of the Ei, i = 1,...,6
are as follows:
'4'i = - € s c -8nT ;
1 2 3 i j
f+ 1 if spacetime indices run over {0,1,2,3),
if spacetime indices run over {1,2.3.4);.
1/2 '6"i jkl
= [-g] sign(ijkl),
where g m det[g..] and (ijkl) is a permutation in S4. 1 J
The parameters W1, W2, W3 of [I251 are defined by
W1.signature(g) = +2; ~ 2 ~ i ~ f jkl = v j; kl - v j; lky
W3Gij = amij; W~w3Ri j = llm . im j
From these definitions it is straightforward
relations relating these two sets of parameters:
E: = wl; 1
€ = -w,; 2
€ = w2w,; 3
E4 = W1,
to find the
To transform tensor equations in one notation system
(barred system) to another system (unbarred) it is necessary to
calculate the six transformation parameters a i = 1 , . . . , 6 by i '
the relation
a = E 2 i = 1 , . . . , 6 . i i i
With the parameters ai we can convert tensor equations from one
notation system to the other by the use of the following -
equations:
- g = g, (g is the determinant of g )
i .I
R i j r l a a R
1 2 i j k l ' (insert one factor a for each index to
1
be raised) - R = a a R
i j 2 3 i j ' (insert one factor a% for each index to be
raised)
- R = a a a R , (R is the curvature scalar)
1 2 3 - G = a a G (insert one factor a for each index to be
i j 2 3 i j ' 1
raised)
The conformal curvature tensor C i j k l and the projective
curvature tensor W transform in the same way as the Riemann
curvature tensor R, j k l under changes in and c2 but it
misbehaves under changes in E3.
For equations which involve the duals of tensor we use the
transformation format -
All tensor expressions not involving the conformal curvature
tensor or the projective curvature tensor should be transform-
able from one notation system - to another by the above rules,
The transformation scheme just outlined can also be extended to
the tetrad equivalents of the above expressions.
For expressions employing the Cartan differential algebra
we must add another indicator for the possible choices of the \
wedge product. Unfortunately since the different choices of
wedge product are related by the "cocyle identity" 11411 the
indicator does not take just two values. We will omit further
discussion on this and refer the reader to the literature
[141,142].
APPENDIX B
SYMBOLS AND NOTATION
The general naming conventions and usage of symbols
employed in this thesis are listed below. Lower case Greek
indices take their values in a specified index set, They will
always be used as labels, usually linking coordinate functions
to their associated coordinate chart. They will never be used
as tensorial indices. Lower case Roman indices have range
{0,1,2,3) and are used to indicate spacetime components in a
holonomic basis. Upper case Roman indices have range {0,1,2,3)
and are used to indicate tetrad components for an anholonomic
basis. We also follow the rules:
(a) The Einstein summation convention holds for the following
index types:
(i)-\lower case Roman indices have range {0,1,2,3);
(ii) upper case Roman indices have range {0,1,2,3);
(b) covariant derivatives are indicated by a semi-colon: Xi;k;
(c) partial derivatives are indicated by a comma: Xi,k.
The following list describes most of the symbols used in
this work:
A i acceleration 4-vector
I3 velocity parameter (v/c) b abstract index lowering operator (vectors to forms)
abstract index raising operator (forms to vectors)
space of smooth curves in M
space of n-fold differentiable mappings from M to N
space of smooth mappings from M to N
conformal curvature tensor
covariant derivative with respect to X
Kronecker tensor
i-th coordinate basis vector for the chart (9kat'p,)
frame vectors i = 0, 1, 2, 3.
indicator of ei i.e. ciei8ei = 1
pull-back by f
push-forward by f when f is a local diffeomorphism
components of the - metric tensor in a holonomic basis
detg j t the determinant of gij
components of G in a holonomic basis
Christoffel symbol of the second kind
projection tensor associated with g and ui i j
conformal collineation tensor
coefficient of shear viscosity
Levi-Civita twisted 4-form
coordinate transition map from ( UQ, q B ) to ( U=,cp,)
identity map on a manifold M
chronological future set of p
chronological past set of p
index of the metric tensor g (1)
j N inclusion map for a submanifold N
J+(P) causal future set of p
J-(PI causal past set of p
Lor(M) space of Lorentz metrics on M
4 Lie derivative with respect to X
space-time manifold
type 1 Lorentzian warped product of M, H
Minkowski space-time
energy density
the nullity of the metric tensor ( 0 )
6-neighbourhood of g in fine 6-topology on Lor(M)
thermodynamic pressure
momentum 4-vector -
scalar curvature ,
components of Riemann curvature in a holonomic basis
components of Ricci curvature in a holonomic basis
the signature of the metric tensor g ( + 2 )
shear scalar
shear tensor
stress-energy-momentum tensor
the space of type (r,s) tensors on the vector space V
tangent bundle of M
cotangent space of M at p
tangent space of M at p
spatial expansion rate
8 rate of strain tensor i j
U unit velocity 4-vector i
r coefficient of bulk viscosity
O i j vorticity tensor
[x,yl bracket of vector fields X, Y
APPENDIX C
CALCULATIONS IN NN-COORDINATES
This appendix contains some of the calculations which
were not used directly in the text. They are all computed in
NN-coordinates with metric given by ( 4 . 1 ) . Unless otherwise
noted all velocity dependent calculations used a generalized
comoving velocity as defined in Chapter 4 . The geodesic
equations are
rrV# zsinz8/(2fz) + rr v 8'z/(2fz) + 2u8'fu/f + u" = 0,
where the prime indicate differentiation with respect to an
affine parameter. The conservation equations for a perfect
fluid with a generalized comoving velocity in NN-coordinates
may be written
We now calculate equations for a spacelike collineation
vector in NN-coordinates. We assume that the velocity is
generalized comoving and the collineation vector is parallel to
the spacelike anisotropy vector S . The direction of anisotropy
is assumed orthogonal to the comoving velocity thus has
nonzero components S o = f, S 1 = -f. The spacelike conformal
collineation vector has nonzero components 5 o = fX, = -fl,
where X is the scaling factor. The equations for a conformal
collineation with H i j as collineation tensor are
where V is the conformal factor. Note we set H i j i k = 0 as the
condition for the existence - of a conformal collineation vector.
2
H 2 2 ; 0 = - r X r f / f + rXr f / f 2 + rXr / f - r h r / f + rr X / f
U U v u U U v u u U
- rr X / f - 2r2yu - X ( r I 2 / f + X r r / f , v U U U v
H 2 2 ; 1
= -rlr f / f 2 + rXr f / f 2 + rXruv/f - rlr / f + rr A / f U v v v v v u v
- rr X / f - 2 r Z y + X ( r ) '/f - X r r / f , v V v v U v
H 33; 0
= - r l s in28r f / f 2 + rXsin28r f / f 2 + rks in28ruu/ f U u v U
2 - r l s i n 8 r v u / f + r s in28r A / f - r s in28r X / f U u v u
- 2rZsin2€hp u - Xsin28(r - ) ' / f t Xsin28rurv/ f , u
H 2
33; 1 = -rXsin20r f v / f + rXsin28r f / f 2 + rXsin20ruv/f
u v v 2 - r l s i n 8 rvv / f + r s in28r A '/f - r s in20r A / f u v v v
- 2 r 2 s i n 2 ~ v + Xsin28(r ) 2 / f - Xsin28rUrV/f . v
We</'set y = 0 as t h e cond i t ion f o r a s p e c i a l conformal , ; i j
c o l l i n e a t i o n v e c t o r .
Now we w r i t e t h e equat ions f o r a " t i l t i n g " t i m e l i k e
collineation vector. We assume 5 has nonzero components given
by co = a, E l = b.
The tilting conformal collineation tensor equations are
We set H i j i r = 0 as the condition for the existence of a
conformal timelike tilting collineation vector,
3
H22;o = 2arr fu/f - arrvu
v /f2 + arurv/f2 + 2brr u fu/f3
2 - brr uu /f + b(rUl2/f2 - raurv/f2 - rb u r u /f2 2 - 2r tVu,
H22; 1 = 2arr f /f3 - arrvv
v v /f2 + a(rVl2/f2 + 2brr u f v /f3
2 - brr /f2 + br r /f2 - ravrv/f - rb r /f2 - 2r2vV, uv U v v u
3 H33;0 = 2arsin28 rvfu/f - arsin28rvu/f2 + asin28 rurv/f
2 + 2brsinz8 rufu/f3 - brsin Bruu/f + bsin28(r u 1 '/f
3 2 H33; 1 = 2arsin28r v fv/f - arsin 8rvv/f + asin28(r v ) '/f
2 + 2brsin28r u f v /f3 - brsin 8ruv/f + bbsin2€J r u r v /f2 2 - rsin2e avrv/f - rsin28bvru/f2 - 2r2sin2w v .
The nonzero components - of a timelike conformal
collineation vector parallel to the generalized comoving
velocity are 5, = fl, El = fX. The equations for a timelike
conformal collineation vector parallel to the comoving velocity
We set = 0 as the condition for the existence of a
conformal collineation vector.
H l l ; o = 2fX - 2Xf t 2f X - 2f X , v u V U U v v U
H 11 ; l
= 2fX - 2Af t 8X(f ) '/f - 8f X v v v v v v v 9
H 2
1 2 ; 2 = r h r f / f - rXr f / ( 2 f 2 ) - r X r f / ( 2 f 2 ) - rr X / f
U v v U - v v u v
- rr X / ( 2 f ) - rr X / ( 2 f ) t X ( r ) 2 / f t X r r / f , v U v v v u v
H 2
13; 3 = rXsin28r u f v / f - rXsin28r v f u / ( 2 f 2 ) - rXsin28r v f v / ( 2 f 2 )
2 - r s i n 8r A / f - r s i n 2 8 r X / ( 2 f ) - rs in28r X / ( 2 f ) U v v U v v
, t Xsin28(r ) '/f t Xsin28rurv/ f , v
2 H 2 2 ; 0
= r X r f / f t r h r f / f 2 - r X r f - r r / f - rr X / f u u v u U U V U u U
- rr X / f - 2 r Z y U t A ( r )'/f t Xrr / f , v u u U V
H 2
2 2 ; 1 = r h r f / f + rXr f / f 2 - rXr / f - r X r / f - rr X / f
u v v v u v v v u v
- rr A / f - 2 r Z v V t X ( r ) 2 / f t Xrr / f , v v v U v
2
H,,;o = rXsinZ8r u f u / f t rXsin28rvfu/f - r l s i n 8 r u u / f 2 - rXsin 8 rvu / f - rs in28r X / f - r s in28r X / f
u u V u
- 2 r 2 s i n 2 w u t Xsin28(r u 12 / f t Xsin28rurv/ f ,
APPENDIX D
A MUTENSOR PROGRAM FOR A PERFECT FLUID IN NN-COORDINATES
The following is a MuTensor script file listing which
computes many of the quantities used for General Relativity.
The following program listing was used, together with
variants, to produce many of the calculations used in this
thesis. The programs were developed for the MuTensor computer
algebra system version 3.75 C1431.
%COMMENT
TITLE: PERFSTAN.NNC
STATUS: working LAST UPDATE: 16:50/21/3/88
NOTATION: W=(+,+,+) E=(+,-,+,+,+,+)
This muTensor script computes the field equations in double
null coordinates for a spherically symmetric metric with a
perfect fluid stress-energy tensor the velocity vector is
assumed to be "COMOVING."
A = acceleration vector
G = Einstein tensor (from RIC) W(3)=-1
M = conservation vector from twice contracted Bianchi identity
NG = Einstein tensor (from NRIC) W(3)=+1
NRIC = Ricci tensor (R contracted on 1-3 positions)
P = projection tensor onto 3-space orthogonal to U
R = Riemann tensor E(2)=-1 W(2)=+1
RIC = Ricci tensor (R contracted on 1-4 positions) E(3)=-1
T = stress-energy tensor of type (0,2)
TUP = stress-energy tensor of type (2,O)
U = 4-velocity of fluid
V relative acceleration tensor for 3-space orthogonal to U
W = vorticity tensor of the relative velocity in 3-space
orthogonal to U
VFE = field equation tensor (identically zero) (W(3)=+1)
f = metric component function
m = density of rest mass
p = pressure
r = metric component function
Th = expansion tensor
Units are chosen so that c=h=kappa*G=l.
%
RECLAIM();
DE&NUM:NUMNUM:DENDEN:EXPBAS:BASEXP:~~$
NUMDEN: O$
TRGSQ: 4$
COORDS : '(u v th ph)$
ds: -4*fn2*d(u)*d(v) + rn2*d(th)"2 + rA2*SIN(th)"2*d(ph)"2;
DEPENDS (f(u,v))$
DEPENDS (r(u,v))$
METRIC(ds);
%COMMENT
We compute the Christoffel symbols and read in the velocity
vector so that we can later compute the stress-energy tensor
after we compute the shear tensor etc.
%
CHRISTOFFELl()$
CHRISTOFFELB()$
MKTNSR ('U, '(-I), ' ( ) ) $
U[O] :: f$
U[1] :: f$
U[21 :: O$
U[3] :: O$
%COMMENT
Here we are going to compute the acceleration 4-vector
%
MKTNSR ('A, ( - 1 ' 0 ) s
%COMMENT Here we are going to define the projection tensor \
related to metric and the 4-velocity.
MKTNSR ('P, '(-1 -11, '((1 1 2)))$
%COMMENT Here we define the relative acceleration tensor which
is defined to be the covariant derivative of the 4-velocity
corrected for acceleration orthogonal to the 3-space defined by
the 4-velocity.
%
MKTNSR ('V, '(-1 -1)' '((1 1 2 ) ) ) $
V[a,b] :: U[a,: :b] + A[a]*U[b]$ %COMMENT
Here we decompose the covariant derivative of the 4-velocity
into its symmetric and anti-symmetric parts.
%
MKTNSR ('W, '(-1 -I), '((-1 1 2)))$
Here we are going to compute the shear rate tensor
We also compute the expansion "e".
MKTNSR ('E, '(-1 -1)' '((1 1 2 ) ) ) $
DEPENDS (e(u,v))$
Here we compute the stress-energy momentum tensor.
MKTNSR ('T, '(-1 -I), '((1 1 2)))$
MKTNSR ('TUP, '(1 I ) , '((1 1 2)))$
DEPENDS (p(u,v))$
DEPENDS (m(u,v))$
SHIFT (TKAa,bl);
MKTNSR ('M, '(I), ' ( ) ) $
TUP[a,b] :: T[̂ a,̂ b]$
MCa] :: TUP[a,b,::b]$
RIEMANNO;
WEYL( ) ;
RICCI ( ) ;
MKTNSR ('NRIC, '(-1 '((1 1 2)))s
NRIC[a,bl :: -RIC[a,b]$
%COMMENT
Here we program around a bug in the MuTensor program. It does
not compute the Ricci scalar or the Einstein tensor properly.
%
RR :: NRICCAa,a];
DEPENDS (RR(u,v))$
MKTNSR ('N, '(I), ' ( ) ) $
MKTNSR ('NUP, ( 1 1 '((1 1 2)))$
The vector N should be identically 0 so that G ~ ' a 0. For the ; j
code above this identity is true.
%
N[l;
MKTNSR ('VFE, 1 - 1 ) '((1 1 2)))$
VFE[a,b] :: NG[a,b] - T[a,bl$ SHIFT (VFECAa,b]);
RECLAIM 0;
Several lines for printing output omitted here.
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